* *Rotational Motion

*Rotational Motion*

In This Chapter:

✔ *Angular Measure*

✔ *Angular Velocity*

✔ *Angular Acceleration*

✔ *Moment of Inertia*

✔ *T**orque*

✔ *Rotational Energy and Work*

✔ *Angular Momentum*

__Angular Measure__

In everyday life, angles are measured in degrees, where 360º equals a full turn. A more suitable unit for technical purposes is the *radian *(rad). If a circle is drawn whose center is at the vertex of a particular angle (Fig-ure 7-1), the angle q (Greek letter *theta*) in radians is equal to the ratio between the arc *s *cut by the angle and the radius *r *of the circle:

Figure 7-1

Because the circumference of a circle of radius *r *is 2p*r*, there are 2p rad in a complete revolution (rev). Hence

1 rev = 360° = 2π rad

and so,

1° = 0.01745 rad 1 rad = 57.30°

__Angular Velocity__

__Angular Velocity__

The *angular velocity *of a body describes how fast it is turning about an axis. If a body turns through the angle q in the time *t*, its angular veloci- ty w (Greek letter *omega*) is

Angular velocity is usually expressed in radians per second (rad/s), revolutions per second (rev/s or rps), and revolutions per minute (rev/min or rpm), where

1 rev/s = 2π rad/s = 6.28 rad/s

1 rev/min =(2π /60) rad/s = 0.105 rad/s

The linear velocity *v *of a particle that moves in a circle of radius *r *with the uniform angular velocity w is given by

*v *= ω*r*

Linear velocity = (angular velocity)(radius of circle)

This formula is valid only when *w *is expressed in radian measure.

###### Angular Acceleration

A rotating body whose angular velocity changes from ω0 to ω*f *in the time interval *t *has the *angular acceleration **a *(Greek letter *alpha*) of

A positive value of *a *means that the angular velocity is increasing; a negative value means that it is decreasing. Only constant angular accelerations are considered here.

The formulas relating the angular displacement, velocity, and acceleration of a rotating body under constant angular acceleration are analogous to the formulas relating linear displacement, velocity, and acceleration. If a body has the initial angular velocity *w*0, its angular velocity ω*f*

after a time *t *during which its angular acceleration is a will be

ω*f *= ω*o *+ *a**t*

and, in this time, it will have turned through an angular displacement of

θ= ω_{o}* ^{t }*+ (1/2)

*a*

*t*

^{2}

A relationship that does not involve the time *t *directly is sometimes useful:

ω* _{f}^{2} *= ω

*+ 2*

_{o}^{2}*aθ*

**Solved Problem 7**.**1 **A phonograph turntable initially rotating at 3.5 rad/s makes three complete turns before coming to a stop. (*a*) What is its angular acceleration? (*b*) How much time does it take to come to a stop?

Solution.

(a) The angle in radians that corresponds to 3 rev is

__Moment of Inertia__

The rotational analog of mass is a quantity called *moment of inertia*. The greater the moment of inertia of a body, the greater its resistance to a change in its angular velocity.

##### You Need to Know

###### The value of the moment of inertia I of a body about a particular axis of rotation depends not only upon the body’s mass but also upon how the mass is distributed about the axis.

Let us imagine a rigid body divided into a great many small particles whose masses are *m*1, *m*2, *m*3, … and whose distances from the axis of rotation are respectively *r*1, *r*2, *r*3, … (Figure 7-2).

The moment of inertia of this body is given by

*I *= *m*_{1}*r*_{1} ^{2}+ *m*_{2}*r*_{2} ^{2}+ *m*_{3}*r*_{3} ^{2}+ L = Σ*m**r ^{2}*

where the symbol S (Greek capital letter *sigma*) means “sum of ” as be- fore. The farther a particle is from the axis of rotation, the more it con- tributes to the moment of inertia. The units of *I *are kg · m2 and slug · ft2. Some examples of moments of inertia of bodies of mass *M *are shown in Figure 7-3.

__Torque__

The *torque *t (Greek letter *tau*) exerted by a force on a body is a measure of its effectiveness in turning the body about a certain pivot point. The *moment arm *of a force **F **about a pivot point *O *is the perpendicular distance *L *between the line of action of the force and

*O *(Figure 7-4). The torque τ exerted by the force about *O *has the magnitude

**Figure 7-4 **Four directions along which a force **F **can be applied to a wrench. In (*a*) the moment arm *L *is longest, hence the torque t = *FL *is a maximum. In (*d *) the line of action of **F **passes through the pivot *O*, so *L *= 0 and t = 0.

The torque exerted by a force is also known as the *moment *of the force. A force whose line of action passes through *O *produces no torque about *O *because its moment arm is zero.

Torque plays the same role in rotational motion that force plays in linear motion. A net force *F *acting on a body of mass *m *causes it to un- dergo the linear acceleration *a *in accordance with Newton’s second law of motion *F *= *ma*. Similarly, a net torque t acting on a body of moment of inertia *I *causes it to undergo the angular acceleration a (in rad/s2) in accordance with the formula

*t *= *I**a*

Torque= (moment of inertia) (angular acceleration)

__Remember__

__Remember__

###### In the SI system, the unit of torque is newton · meter (N·m); in the British system, it is the pound · foot (lb·ft).

__Rotational Energy and Work__

The kinetic energy of a body of moment of inertia *I *whose angular ve- locity is w (in rad/s) is

*KE *=( 1/2 )*Iω*^{2}

Kinetic energy =( 1/2 )(moment of inertia) (angular velocity)^{2}

The work done by a constant torque τ that acts on a body while it experiences the angular displacement θ rad is

*W *= *π ω*

Work= (torque) (angular displacement)

The rate at which work is being done when a torque τ acts on a body that rotates at the constant angular velocity w (rad/s) is

*P *= π*ω*

Power= (torque) (angular velocity)

__Angular Momentum__

The equivalent of linear momentum in rotational motion is *angular momentum*. The angular momentum **L **of a rotating body has the magnitude

*L *= *I**ω*

Angular momentum = (moment of inertia) (angular velocity)

The greater the angular momentum of a spinning object, such as a top, the greater its tendency to spin.

Like linear momentum, angular momentum is a vector quantity with direction as well as magnitude. The direction of the angular momentum of a rotating body is given by the right-hand rule (Figure 7-5):

Figure 7-5

When the ﬁngers of the right hand are curled in the direction of rotation, the thumb points in the direction of **L**.

According to the principle of *conservation of angular momentum*, the total angular momentum of a system of bodies remains constant in the absence of a net torque regardless of what happens within the system. Be- cause angular momentum is a vector quantity, its conservation implies that the direction of the axis of rotation tends to remain unchanged.

Table 7.1 compares linear and angular quantities.