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

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. Reason: there is only ONE unique direction that can be used – the rotation axis.

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

The new model is a rigid object under constant angular acceleration Analogous to the particle under constant acceleration

model

2

2 2

1

2

2

1

2

f i

f i i

f i f i

f i i f

t

t t

t

all with consant

Rotational Kinematic Equations

v = u + at

xxff= = ½(½(vvi i + v+ vf f )t)t

vv22 = u = u22 + 2as + 2as

s = ut + ½ at2

All for constant

Comparison Between Rotational and Linear Equations

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

s r

v r

a r

Speed Comparison The linear velocity is

always tangent to the circular path Called the tangential

velocity

The magnitude is defined by the tangential speed

ds dv r r

dt dt

All points on the rigid object will have the same angular speed, but not the same tangential

speed

Acceleration Comparison The tangential

acceleration is the derivative of the tangential velocity

t

dv da r r

dt dt

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

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 acceleration2

2C

va r

r since v = r

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

2 2 2 2 2 4 2 4t ra a a r r r

t

dv da r r

dt dt

2

2C

va r

r

Rotational Motion Example For a compact disc player to read a CD, the angular speed must

vary to keep the tangential speed constant (vt = r) At the inner sections, the angular speed is faster than at the

outer sections

The tangential speed of the disc surface at the location of

the lens is constant. This requires, according to v = r, that the angular speed vary

as the laser-lens system moves radially along the disc.

In a typical compact disc player, the constant speed of the surface at the point of the laser lens-system is 1.3 m/s.

a. Find the angular speed of the disc in revolutions per minute when information is being read from the innermost first track (r = 23 mm) and the outermost final track (r = 58 mm).

i = v/ri = 1.3/2.3 x 10-2 = 57 s-1 x (1/2)(60/1) = 540 rev/min f = v/rf = 1.3/5.8 x 10-2 = 22 s-1 x (1/2)(60/1) = 210 rev/min

The player adjusts the angular speed of the disc between these two values in order to maintain a tangential speed of 1.3 m/s at the lens – so that the information passes at a constant rateb. The maximum playing time of a standard music CD is 74 min and 33s. How many revolutions does the disc make during that time? =½(i + f)t = ½(57 + 22)(4473) = 1.8 x 105 rad x (1/2) = 2.8 x 104

revc. What is the total length the track moves past the objective lens during this time?xf = vit = 1.3 x 4473 = 5.8 x 103 m = 5.8

kmd. What is the angular acceleration of the CD over the 4473 s time interval? Assume that is constant.

= (f - i)/t = (22-57)(4473) = -7.8 x 10-3 rad s-2

Disc undergoes very gradual decrease in its rotation rate (since the disc has a higher angular speed at the beginning, ri)

Rotational Kinetic Energy

An object rotating about some axis with an angular speed, ω, has rotational kinetic energy even though it may not have any translational kinetic energy

Each particle has a kinetic energy of Ki = ½ mivi

2

Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute vi = i r

Ki = ½mi2ri2

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

2 2

2 2 2

1

2

1 1

2 2

R i i ii i

R i ii

K K m r

K m r I

There is an analogy between the kinetic energies associated with linear motion (K = ½ mv 2) and the kinetic energy associated with rotational motion (KR= ½ 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)

Analogous to mass, but not an intrinsic property – depends

on choice of rotation axis

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

2i i

i

I r m

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

lim 2 20im i i

i

I r m r dm

2I r dV

Notes on Various Densities

Volumetric Mass Density → mass per unit volume: = m / V

Surface Mass Density → mass per unit thickness of a sheet of uniform thickness, t : t

Linear Mass Density → mass per unit length of a rod of uniform cross-sectional area: = m / L =

Moment of Inertia of a Uniform Rigid Rod

The shaded area has a mass dm = dx

Then the moment of inertia is

/ 22 2

/ 2

21

12

L

y L

MI r dm x dx

L

I ML

= (M/L)dx

= (M/L)∫-L/2 x2dx

= (M/L)[x3/3]-L/2

=(M/3L) [L3/8 –(-L3/8)]

= (M/3L)(L3/4) = ML2/12

L/2

L/2

Moment of Inertia of a Uniform Solid Cylinder Divide the cylinder into

concentric shells with radius r, thickness dr and length L

dm = dV = 2Lr dr Then for I

2 2

2

2

1

2

z

z

I r dm r Lr dr

I MR

Area of ring is dA = 2r.dr

dV = LdA

= L 2r.dr

= 2L∫0 r3dr

= 2L[r4/4]0

= ½LR4

Total volume is R2L so = M/V = M/R2L

M

I = ½MR2

R

R