Chapter 10:

Rotation of a Rigid

Object About a fixed

Axis.

• Exam # 3 Nov 21

• Study: Problems from Exam # 1 and Exam #2,

• chap. #7,

• conservation of energy (powerpoint),

• chp. # 9 (9.1 – 9.6),

• chp. # 10.1 – 10.3

November 15, 2013

Rotational Motion

• Angular Position and Radian

• Angular Velocity

• Angular Acceleration

• Rigid Object under Constant

Angular Acceleration

• Angular and Translational

Quantities

• Rotational Kinetic Energy

• Moments of Inertia

November 15, 2013

Angle and Radian • What is the circumference S ?

• q can be defined as the arc length s along a circle divided by the radius r:

• q is a pure number, but commonly is given the artificial unit, radian (“rad”)

r

q s

rq

rs )2( r

s2

s

q Whenever using rotational equations, you must use angles expressed in radians

November 15, 2013

Conversions • Comparing degrees and radians

• Converting from degrees to radians

• Converting from radians to degrees

180)( rad

3601 57.3

2rad

180

rad degrees

q q

360)(2 rad

)(180

)(deg radrees q

q

November 15, 2013

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

• This simplification allows analysis of the motion of an

extended object

November 15, 2013

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

• As the particle moves, the only

coordinate that changes is q

• As the particle moves through q, it

moves though an arc length s.

• The angle q, measured in radians, is

called the angular position.

November 15, 2013

Displacement • Displacement is a change of position in time.

• Displacement:

• f stands for final and i stands for initial.

• It is a vector quantity.

• It has both magnitude and direction: + or - sign

• It has units of [length]: meters.

)()( iiff txtxx

x1 (t1) = + 2.5 m x2 (t2) = - 2.0 m Δx = -2.0 m - 2.5 m = -4.5 m

x1 (t1) = - 3.0 m x2 (t2) = + 1.0 m Δx = +1.0 m + 3.0 m = +4.0 m

November 15, 2013

Angular Displacement

• The angular displacement is

defined as the angle the

object rotates through during

some time interval

• SI unit: radian (rad)

• This is the angle that the

reference line of length r

sweeps out

f iq q q

November 15, 2013

Velocity

• Velocity is the rate of change of position.

• Velocity is a vector quantity.

• Velocity has both magnitude and direction.

• Velocity has a unit of [length/time]: meter/second.

• Definition:

• Average velocity

• Average speed

• Instantaneous

velocity

avg

total distances

t

0lim

t

x dxv

t dt

t

xx

t

xv

if

avg

November 15, 2013

Average and Instantaneous

Angular Speed

• The average angular speed, ωavg, of a rotating rigid object is

the ratio of the angular displacement to the time interval

• The instantaneous angular speed is defined as the limit of the

average speed as the time interval approaches zero

• SI unit: radian per second (rad/s)

• Angular speed positive if rotating in counterclockwise

• Angular speed will be negative if rotating in clockwise

f iavg

f it t t

q q q

lim

0 t

d

t dt

q q

November 15, 2013

Average Angular Acceleration

• The average angular acceleration, a, 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:

f i

avg

f it t t

a

t = ti: i t = tf: f

November 15, 2013

Instantaneous Angular Acceleration

• The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0

• SI Units of angular acceleration: rad/s²

• Positive angular acceleration is in the counterclockwise. • if an object rotating counterclockwise is speeding up

• if an object rotating clockwise is slowing down

• Negative angular acceleration is in the clockwise.

• if an object rotating counterclockwise is slowing down

• if an object rotating clockwise is speeding up

lim

0 t

d

t dt

a

November 15, 2013

Rotational Kinematics • A number of parallels exist between the equations for

rotational motion and those for linear motion.

• 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

t

x

tt

xxv

if

if

avg

f iavg

f it t t

q q q

November 15, 2013

Comparison Between

Rotational and Linear Equations

November 15, 2013

Example 1

• A wheel rotates with a constant angular acceleration of 5.0 rad/s2. If the angular speed of the wheel is 1.5 rad/s at t = 0

(a) through what angle does the wheel rotate between

t = 0 and t = 2.0 s? Given your answer in radians and in revolutions.

(b) What is the angular speed of the wheel at t = 2.0 s?

November 15, 2013

Example 2

• During a certain time interval, the angular position of a swinging

• door is described by , where θ is in radians

• and t is in seconds. Determine the angular position, angular speed,

• and angular acceleration of the door at the following times.

• • a) t= 0 s

• • b) t = 3.00 s

200.20.1000.5 tt q

November 15, 2013

Example 3

A rotating wheel requires 3.0 s to rotate through 37.0 revolutions. Its angular

speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant

angular acceleration of the wheel?

November 15, 2013

Relationship Between Angular and

Linear Quantities

• Every point on the rotating object has

the same angular motion

• Every point on the rotating object

does not have the same linear motion

• Displacement

• Speeds

• Accelerations

s rq

v r

a ra

November 15, 2013

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

rvorr

v arat

November 15, 2013

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

222 )(

rr

r

r

var

November 15, 2013

Home Example 4

A car accelerates uniformly from rest and reaches a speed of 22.0 m/s in

9.00 s. If the diameter of a tire is 58.0 cm, find (a) the number of

revolutions the tire makes during this motion, assuming that no slipping

occurs. (b) What is the final angular speed of a tire in revolutions per

second?

November 15, 2013

Moment of Inertia of Point Mass

• For a single particle, the definition of moment of inertia is

• m is the mass of the single particle

• r is the rotational radius

• SI units of moment of inertia are kg.m2

• Moment of inertia and mass of an object are different quantities

• It depends on both the quantity of matter and its distribution (through the r2 term)

2mrI

November 15, 2013

Moment of Inertia of Point Mass

• For a composite particle, the definition of moment of inertia

is

• mi is the mass of the ith single particle

• ri is the rotational radius of ith particle

• SI units of moment of inertia are kg.m2

• Consider an unusual baton made up of four sphere fastened to

the ends of very light rods

• Find I about an axis perpendicular to the page and passing

through the point O where the rods cross

...2

44

2

33

2

22

2

11

2 rmrmrmrmrmI ii

222222222 mbMaMambMambrmI ii

November 15, 2013

Example: Moment of Inertia

of a Uniform Rigid Rod

• The shaded area has a

mass

• dm = l dx

• Then the moment of

inertia is

/ 2

2 2

/ 2

21

12

L

yL

MI r dm x dx

L

I ML

November 15, 2013

Moment of Inertia for some other common

shapes

November 15, 2013