14 - rotation of rigid bodies.pptx
TRANSCRIPT
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Rotation of Rigid Bodies
Ms. Mikaela Irene Fudolig
Physics 71 Lecture
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Rigid Bodies and Rotation
Rigid bodiesbodies with a definite size and shapedo not deform
Rotationfor now, we consider rotation about a fixed axis
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Example 1
Identify the axis of rotation for each of the following rotating objects:
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Describing Rotational Motion
Use polar coordinates, (r,)!
s r is in
RADIANS!
Signs of :
+ if rotation is CCW
– if rotation is CLOCKWISE
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Exercise 1
A girl is on a merry-go-round, 5 meters from the center of the circular platform. What is her angular displacement if she travels a total distance of 2 meters after some time t?
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Angular velocity,
Analogous to the linear velocity, vRate of change in angular displacement
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Angular velocity,
Average angular velocityInstantaneous angular velocityAngular speed
magnitude of the angular velocity
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Average Angular Velocity
The average angular velocity over a time interval t = tf - ti is
ˆ av kt
*assumes
that the rotation axis is the z-axis.*
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(Instantaneous) Angular velocity
rate of change of the angular displacement
VECTOR!
ˆ d
kdt
*assumes
that the rotation axis is the z-axis.*
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Angular velocity,
Direction along the axis:UPWARDS (+), if
rotation is CCWDOWNWARDS (-), if
rotation is CLOCKWISE
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Angular velocity,
CAUTION:
“Clockwise” and “CCW” are NOT directions for the ANGULAR VELOCITY!!!
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Angular speed,
magnitude of the angular velocity
d
dt
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Angular acceleration,
Analogous to the linear acceleration, aHow fast the angular velocity changes
2
2
ˆ
ˆ
dk
dt
dk
dt
*assumes that the rotation axis is the z-axis.*
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Angular acceleration,
Direction along the axis:SAME DIRECTION
AS , if rotation SPEEDS UP
OPPOSITE DIRECTION AS , if rotation SLOWS DOWN
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Exercise 2
You put a DVD in the DVD drive of your laptop, and it starts rotating clockwise.Draw a diagram of the rotating DVD, indicating
direction of rotation.Draw the direction of the angular velocity
vector.Draw the direction of the angular acceleration
vector.
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Linear vs. Angular Quantities
Linear Angular
x,s
v
a
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Motion with Constant Angular Acceleration
2 20
0
2
2t
Analogous to motion with constant linear acceleration (aka Chapter 2)
0
20
1
2
t
t t
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Exercise 3
Y&F ex.9.3. A DVD is slowing to a stop. At t=0, its angular velocity is 25.7rad/s and its angular acceleration is a constant –10.0rad/s2.What is the disc’s angular velocity at
t=0.300s?What angle does the line PQ make with the
+x-axis at t=0.300s (I.e., what is (t=0.300s)?
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Linear Kinematics of a rotating particle
A particle in a rotating body moves in a circle:
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Linear/tangential velocity of a rotating particle
From s=r, we can get the expression for the linear speed of a rotating particle a constant distance r from the axis of rotation:
v rThe direction of the linear velocity vector is TANGENT TO ITS CIRCULAR PATH, parallel to the infinitesimal displacements!
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Acceleration of a rotating particle
Tangentialchange in linear speed
Radialchange in direction of the linear velocity
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Tangential acceleration
The tangential acceleration of a rotating particle a distance r from the axis of rotation is:
tan
da r r
dt
Direction: parallel or antiparallel to the linear velocity
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Radial acceleration
The radial acceleration of a rotating particle a distance r from the axis of rotation is:
22
rad
va r
r
Direction: pointing towards the axis of rotation
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Exercise 4Y&F 9.26. An electric ceiling fan with
blades 0.750m in diameter is rotating with constant angular acceleration 0.900rev/s2 about a fixed axis with an initial angular velocity of 0.250rev/s. After 2.00s:what is the angular velocity of the fan?what is the linear speed of the tip of a fan
blade?
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Exercise 4
Y&F 9.26. A electric ceiling fan with blades 0.750m in diameter is rotating with constant angular acceleration 0.900rev/s2 about a fixed axis with an initial angular velocity of 0.250rev/s. After 2.00s:what is the radial acceleration?what is the tangential acceleration?