chapter 10 rotational motion 2 3 10.1 rigid object a rigid object is one that is nondeformable the...
TRANSCRIPT
![Page 1: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/1.jpg)
Chapter 10
Rotational Motion
![Page 2: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/2.jpg)
2
![Page 3: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/3.jpg)
3
10.1 Rigid Object A rigid object is one that is
nondeformable The relative locations of all particles
making up the object remain constant All real objects are deformable to some
extent, but the rigid object model is very useful in many situations where the deformation is negligible
![Page 4: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/4.jpg)
4
Angular Position Axis of rotation is
the center of the disc
Choose a fixed reference line
Point P is at a fixed distance r from the origin
Fig 10.1
![Page 5: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/5.jpg)
5
Angular Position, 2 Point P will rotate about the origin in a circle
of radius r Every particle on the disc undergoes circular
motion about the origin, O Polar coordinates are convenient to use to
represent the position of P (or any other point)
P is located at (r, ) where r is the distance from the origin to P and is the measured counterclockwise from the reference line
![Page 6: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/6.jpg)
6
Angular Position, 3 As the particle
moves, the only coordinate that changes is
As the particle moves through it moves though an arc length s.
The arc length and r are related: s = r
Fig 10.1
![Page 7: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/7.jpg)
7
Radian This can also be expressed as
is a pure number, but commonly is given the artificial unit, radian
One radian is the angle subtended by an arc length equal to the radius of the arc
![Page 8: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/8.jpg)
8
Conversions Comparing degrees and radians
Converting from degrees to radians
![Page 9: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/9.jpg)
9
Angular Position, final We can associate the angle with the entire
rigid object as well as with an individual particle Remember every particle on the object rotates
through the same angle The angular position of the rigid object is the
angle between the reference line on the object and the fixed reference line in space The fixed reference line in space is often the x-
axis
![Page 10: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/10.jpg)
10
Angular Displacement The angular
displacement is defined as the angle the object rotates through during some time interval
This is the angle that the reference line of length r sweeps out
Fig 10.2
![Page 11: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/11.jpg)
11
Average Angular Speed The average angular speed, , of a
rotating rigid object is the ratio of the angular displacement to the time interval
![Page 12: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/12.jpg)
12
Instantaneous Angular Speed The instantaneous angular speed is
defined as the limit of the average speed as the time interval approaches zero
![Page 13: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/13.jpg)
13
Angular Speed, final Units of angular speed are radians/sec
rad/s or s-1 since radians have no dimensions
Angular speed will be positive if is increasing (counterclockwise)
Angular speed will be negative if is decreasing (clockwise)
![Page 14: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/14.jpg)
14
Average Angular Acceleration The average angular acceleration, ,
of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:
![Page 15: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/15.jpg)
15
Instantaneous Angular Acceleration
The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0
Units of angular acceleration are rad/s2 or s-2 since radians have no dimensions
![Page 16: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/16.jpg)
16
Angular Motion, General Notes When a rigid object rotates about a
fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration So all characterize the motion of
the entire rigid object as well as the individual particles in the object
![Page 17: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/17.jpg)
17
Directions, details Strictly speaking, the
speed and acceleration ( are the magnitudes of the velocity and acceleration vectors
The directions are actually given by the right-hand rule
Fig 10.3
![Page 18: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/18.jpg)
18
Directions, final The direction of is along the axis of
rotation By convention, its direction is out of the plane of
the diagram when the rotation is counterclockwise its direction is into of the plane of the diagram
when the rotation is clockwise The direction of is the same as if the
angular speed is increasing and antiparallel if the speed is decreasing
![Page 19: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/19.jpg)
19
10.2 Rotational Kinematics Under constant angular acceleration,
we can describe the motion of the rigid object using a set of kinematic equations These are similar to the kinematic
equations for linear motion The rotational equations have the same
mathematical form as the linear equations
![Page 20: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/20.jpg)
20
Rotational Kinematic Equations
![Page 21: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/21.jpg)
21
Comparison Between Rotational and Linear Equations
![Page 22: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/22.jpg)
22
![Page 23: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/23.jpg)
23
![Page 24: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/24.jpg)
24
10.3 Relationship Between Angular and Linear Quantities Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion
![Page 25: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/25.jpg)
25
Speed Comparison
The linear velocity is always tangent to the circular path called the tangential
velocity The magnitude is
defined by the tangential speed
Fig 10.4
![Page 26: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/26.jpg)
26
• If you can't see the image above, please install Shockwave Flash Player.• If this active figure can’t auto-play, please click right button, then click play.
NEXT
Active Figure10.4
![Page 27: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/27.jpg)
27
Acceleration Comparison
The tangential acceleration is the derivative of the tangential velocity
Fig 10.5
![Page 28: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/28.jpg)
28
Speed and Acceleration Note All points on the rigid object will have the
same angular speed, but not the same tangential speed
All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration
The tangential quantities depend on r, and r is not the same for all points on the object
![Page 29: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/29.jpg)
29
Centripetal Acceleration An object traveling in a circle, even
though it moves with a constant speed, will have an acceleration Therefore, each point on a rotating rigid
object will experience a centripetal acceleration
![Page 30: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/30.jpg)
30
Resultant Acceleration The tangential component of the
acceleration is due to changing speed The centripetal component of the
acceleration is due to changing direction Total acceleration can be found from
these components
![Page 31: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/31.jpg)
31
10.4 Rotational Kinetic Energy An object rotating about some axis with an
angular speed, , has rotational kinetic energy even though it may not have any translational kinetic energy
Each particle has a kinetic energy of Ki = 1/2 mivi
2
Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute vi = i r
![Page 32: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/32.jpg)
32
Fig 10.6
![Page 33: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/33.jpg)
33
Rotational Kinetic Energy, cont The total rotational kinetic energy of the
rigid object is the sum of the energies of all its particles
Where I is called the moment of inertia
![Page 34: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/34.jpg)
34
Rotational Kinetic Energy, final There is an analogy between the kinetic
energies associated with linear motion (K = 1/2 mv 2) and the kinetic energy associated with rotational motion (KR= 1/2 I2)
Rotational kinetic energy is not a new type of energy, the form is different because it is applied to a rotating object
The units of rotational kinetic energy are Joules (J)
![Page 35: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/35.jpg)
35
Moment of Inertia The definition of moment of inertia is
The dimensions of moment of inertia are ML2 and its SI units are kg.m2
We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements, each of mass mi
![Page 36: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/36.jpg)
36
Moment of Inertia, cont We can rewrite the expression for I in terms
of m
With the small volume segment assumption,
If is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known
![Page 37: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/37.jpg)
37
![Page 38: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/38.jpg)
38
![Page 39: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/39.jpg)
39
![Page 40: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/40.jpg)
40
![Page 41: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/41.jpg)
41
![Page 42: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/42.jpg)
42
![Page 43: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/43.jpg)
43
Fig 10.7
![Page 44: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/44.jpg)
44
![Page 45: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/45.jpg)
45
![Page 46: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/46.jpg)
46
![Page 47: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/47.jpg)
47
![Page 48: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/48.jpg)
48
Moment of Inertia of a Uniform Solid Cylinder Divide the cylinder
into concentric shells with radius r, thickness dr and length L
Then for I
Fig 10.8
![Page 49: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/49.jpg)
49
![Page 50: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/50.jpg)
50
![Page 51: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/51.jpg)
51
![Page 52: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/52.jpg)
52
Fig 10.9
![Page 53: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/53.jpg)
53
![Page 54: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/54.jpg)
54
![Page 55: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/55.jpg)
55
![Page 56: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/56.jpg)
56
10.5 Torque Torque, , is the tendency of a force to
rotate an object about some axis Torque is a vector = r F sin = F d
F is the force is the angle the force makes with the
horizontal d is the moment arm (or lever arm)
![Page 57: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/57.jpg)
57
Torque, cont
The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin
Fig 10.10
![Page 58: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/58.jpg)
58
Torque, final The horizontal component of the force
(F cos ) has no tendency to produce a rotation
Torque will have direction If the turning tendency of the force is
counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative
![Page 59: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/59.jpg)
59
Net Torque
The force F1 will tend to cause a counterclockwise rotation about O
The force F2 will tend to cause a clockwise rotation about O
netF1d1 – F2d2
Fig 10.11
![Page 60: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/60.jpg)
60
• If you can't see the image above, please install Shockwave Flash Player.• If this active figure can’t auto-play, please click right button, then click play.
NEXT
Active Figure10.11
![Page 61: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/61.jpg)
61
Torque vs. Force Forces can cause a change in linear
motion Described by Newton’s Second Law
Forces can cause a change in rotational motion The effectiveness of this change depends
on the force and the moment arm The change in rotational motion depends
on the torque
![Page 62: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/62.jpg)
62
Torque Units The SI units of torque are N.m
Although torque is a force multiplied by a distance, it is very different from work and energy
The units for torque are reported in N.m and not changed to Joules
![Page 63: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/63.jpg)
63
Torque as a Vector Product Torque is the vector
product or cross product of two other vectors
Fig 10.12
![Page 64: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/64.jpg)
64
• If you can't see the image above, please install Shockwave Flash Player.• If this active figure can’t auto-play, please click right button, then click play.
NEXT
Active Figure10.12
![Page 65: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/65.jpg)
65
Vector Product, General
Given any two vectors, and
The vector product
is defined as a third vector, whose magnitude is
The direction of C is given by the right-hand rule Fig 10.13
![Page 66: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/66.jpg)
66
Properties of Vector Product The vector product is not commutative
If is parallel ( = 0o or 180o) to
then This means that
If is perpendicular to then
![Page 67: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/67.jpg)
67
Vector Products of Unit Vectors
The signs are interchangeable For example,
![Page 68: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/68.jpg)
68
![Page 69: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/69.jpg)
69
Fig 10.14
![Page 70: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/70.jpg)
70
![Page 71: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/71.jpg)
71
![Page 72: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/72.jpg)
72
![Page 73: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/73.jpg)
73
![Page 74: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/74.jpg)
74
10.6 The Rigid Object In Equilibrium
Fig 10.15 (a) & (b)
![Page 75: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/75.jpg)
75
Problem Solving Strategy – Rigid Object in Equilibrium Conceptualize
Identify the forces acting on the object Think about the effect of each force on the
rotation of the object, if the force was acting by itself
Categorize Confirm the object is a rigid object in
equilibrium
![Page 76: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/76.jpg)
76
Problem Solving Strategy – Rigid Object in Equilibrium, 2 Analyze
Draw a free body diagram Label all external forces acting on the
object Resolve all the forces into rectangular
components Apply the first condition of equilibrium
![Page 77: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/77.jpg)
77
Problem Solving Strategy – Rigid Object in Equilibrium, 3 Analyze, cont
Choose a convenient axis for calculating torques Choose an axis that simplifies your calculations
Apply the second condition of equilibrium Solve the simultaneous equations
Finalize Be sure your results are consistent with the free
body diagram Check calculations
![Page 78: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/78.jpg)
78
![Page 79: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/79.jpg)
79
Fig 10.16(a)
![Page 80: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/80.jpg)
80
![Page 81: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/81.jpg)
81
Fig 10.16(b) & (c)
![Page 82: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/82.jpg)
82
![Page 83: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/83.jpg)
83
![Page 84: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/84.jpg)
84
![Page 85: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/85.jpg)
85
![Page 86: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/86.jpg)
86
![Page 87: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/87.jpg)
87
![Page 88: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/88.jpg)
88
Fig 10.17
![Page 89: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/89.jpg)
89
![Page 90: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/90.jpg)
90
![Page 91: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/91.jpg)
91
![Page 92: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/92.jpg)
92
10.7 Torque and Angular Acceleration on a Particle The magnitude of the torque produced by a force
around the center of the circle is = Ft r = (mat) r
The tangential acceleration is related to the angular acceleration = (mat) r = (mr) r = (mr 2)
Since mr 2 is the moment of inertia of the particle, = I The torque is directly proportional to the angular
acceleration and the constant of proportionality is the moment of inertia
![Page 93: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/93.jpg)
93
![Page 94: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/94.jpg)
94
Fig 10.18(a) & (b)
![Page 95: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/95.jpg)
95
![Page 96: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/96.jpg)
96
![Page 97: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/97.jpg)
97
![Page 98: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/98.jpg)
98
![Page 99: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/99.jpg)
99
![Page 100: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/100.jpg)
100
Work in Rotational Motion Find the work done by a
force on the object as it rotates through an infinitesimal distance ds = r d
The radial component of the force does no work because it is perpendicular to the displacement Fig 10.19
![Page 101: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/101.jpg)
101
Work in Rotational Motion, cont Work is also related to rotational kinetic
energy:
This is the same mathematical form as the work-kinetic energy theorem for translation
If an object is both rotating and translating, W = K + KR
![Page 102: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/102.jpg)
102
Power in Rotational Motion The rate at which work is being done in
a time interval dt is the power
This is analogous to P = Fv in a linear system
![Page 103: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/103.jpg)
103
![Page 104: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/104.jpg)
104
![Page 105: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/105.jpg)
105
Fig 10.20
![Page 106: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/106.jpg)
106
![Page 107: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/107.jpg)
107
![Page 108: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/108.jpg)
108
10.8 Angular Momentum
The instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum
Fig 10.21
![Page 109: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/109.jpg)
109
• If you can't see the image above, please install Shockwave Flash Player.• If this active figure can’t auto-play, please click right button, then click play.
NEXT
Active Figure10.21
![Page 110: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/110.jpg)
110
Torque and Angular Momentum The torque is related to the angular momentum
Similar to the way force is related to linear momentum
This is the rotational analog of Newton’s Second Law The torque and angular momentum must be
measured about the same origin This is valid for any origin fixed in an inertial frame
![Page 111: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/111.jpg)
111
More About Angular Momentum The SI units of angular momentum are
(kg.m2)/ s The axes used to define the torque and
the angular momentum must be the same
When several forces are acting on the object, the net torque must be used
![Page 112: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/112.jpg)
112
Angular Momentum of a System of Particles The total angular momentum of a
system of particles is defined as the vector sum of the angular momenta of the individual particles
Differentiating with respect to time
![Page 113: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/113.jpg)
113
Angular Momentum of a Rotating Rigid Object, cont To find the angular momentum of the
entire object, add the angular momenta of all the individual particles
This is analogous to the translational momentum of p = m v
![Page 114: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/114.jpg)
114
Summary of Useful Equations
![Page 115: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/115.jpg)
115
![Page 116: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/116.jpg)
116
![Page 117: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/117.jpg)
117
![Page 118: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/118.jpg)
118
![Page 119: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/119.jpg)
119
![Page 120: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/120.jpg)
120
10.9 Conservation of Angular Momentum The total angular momentum of a system is
conserved if the resultant external torque acting on the system is zero Net torque = 0 -> means that the system is
isolated For a system of particles, Ltot = Ln = constant
![Page 121: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/121.jpg)
121
Conservation of Angular Momentum, cont If the mass of an isolated system
undergoes redistribution, the moment of inertia changes The conservation of angular momentum
requires a compensating change in the angular velocity
Ii i = If f This holds for rotation about a fixed axis and for rotation
about an axis through the center of mass of a moving system
The net torque must be zero in any case
![Page 122: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/122.jpg)
122
Conservation Law Summary For an isolated system -
(1) Conservation of Energy: Ei = Ef
(2) Conservation of Linear Momentum:
(3) Conservation of Angular Momentum:
![Page 123: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/123.jpg)
123
Gyroscope Angular momentum is the basis of the
operation of a gyroscope A gyroscope is a spinning object used to
control or maintain the orientation in space of the object or a system containing the object
Gyroscopes undergo precessional motion The symmetry axis rotates, sweeping out a
cone
![Page 124: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/124.jpg)
124
Fig 10.22
![Page 125: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/125.jpg)
125
Fig 10.23
![Page 126: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/126.jpg)
126Fig 10.24
![Page 127: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/127.jpg)
127
![Page 128: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/128.jpg)
128
![Page 129: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/129.jpg)
129
![Page 130: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/130.jpg)
130
![Page 131: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/131.jpg)
131
![Page 132: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/132.jpg)
132
![Page 133: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/133.jpg)
133
10.10 Precessional Motion of a Gyroscope
The external forces acting on the top are the normal and the gravitational forces
The torque due to the gravitational force is in the xy plane
Only the direction of the angular momentum changes, causing the precession
Fig 10.25
![Page 134: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/134.jpg)
134
10.11 Rolling Object
The red curve shows the path moved by a point on the rim of the object
This path is called a cycloid The green line shows the path of the center of mass
of the object
Fig 10.26
![Page 135: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/135.jpg)
135
Pure Rolling Motion The surfaces must exert friction forces on
each other Otherwise the object would slide rather than roll
In pure rolling motion, an object rolls without slipping
In such a case, there is a simple relationship between its rotational and translational motions
![Page 136: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/136.jpg)
136
Rolling Object, Center of Mass The velocity of the center of mass is
The acceleration of the center of mass is
![Page 137: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/137.jpg)
137
Rolling Object, Other Points A point on the rim,
P, rotates to various positions such as Q and P ’
At any instant, the point on the rim located at point P is at rest relative to the surface since no slipping occurs
Fig 10.27
![Page 138: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/138.jpg)
138
Total Kinetic Energy of a Rolling Object The total kinetic energy of a rolling
object is the sum of the translational energy of its center of mass and the rotational kinetic energy about its center of mass K = 1/2 ICM 2 + 1/2 MvCM
2
![Page 139: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/139.jpg)
139
Parallel-Axis Theorem For an arbitrary axis, the parallel-axis
theorem often simplifies calculations The theorem states Ip = ICM + MD 2
Ip is about any axis parallel to the axis through the center of mass of the object
ICM is about the axis through the center of mass
D is the distance from the center of mass axis to the arbitrary axis
![Page 140: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/140.jpg)
140
Total Kinetic Energy, Example Accelerated rolling
motion is possible only if friction is present between the sphere and the incline The friction produces
the net torque required for rotation
![Page 141: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/141.jpg)
141
Total Kinetic Energy, Example cont Despite the friction, no loss of mechanical
energy occurs because the contact point is at rest relative to the surface at any instant Let U = 0 at the bottom of the plane Kf + U f = Ki + Ui
Kf = 1/2 (ICM / R 2) vCM2 + 1/2 MvCM
2
Ui = Mgh Uf = Ki = 0
![Page 142: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/142.jpg)
142
![Page 143: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/143.jpg)
143
Fig 10.28
![Page 144: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/144.jpg)
144
![Page 145: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/145.jpg)
145
![Page 146: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/146.jpg)
146
![Page 147: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/147.jpg)
147
• If you can't see the image above, please install Shockwave Flash Player.• If this active figure can’t auto-play, please click right button, then click play.
NEXT
Active Figure10.28
![Page 148: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/148.jpg)
148
10.12 Turning a Spacecraft Here the gyroscope
is not rotating The angular
momentum of the spacecraft about its center of mass is zero
Fig 10.29(a)
![Page 149: Chapter 10 Rotational Motion 2 3 10.1 Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the](https://reader038.vdocument.in/reader038/viewer/2022103006/56649efd5503460f94c10721/html5/thumbnails/149.jpg)
149
Turning a Spacecraft, cont Now assume the
gyroscope is set into motion
The angular momentum must remain zero
The spacecraft will turn in the direction opposite to that of the gyroscope
Fig 10.29(b)