Chapter 10

Rotational Motion and Torque

10.1- Angular Position, Velocity and Acceleration

• For a rigid rotating object a point P will rotate in a circle of radius r away from the axis of rotation.

10.1

• The location of point P can be described in polar coordinates (r , θ).

• The circular distance traveled is called the arc length according to

• When θ is measured in radians(1 radian is the angle swept by an arc length equal

to the radius).

rs

10.1

• Angles measured in radians, degrees, revolutions

2π rad = 360o = 1 rev• Angular displacement- the change in angular

position

if

10.1

• Angular Velocity- the rate of change in angular displacement– For constant rotations or averages

– For Angular Position as function of time

• Measured in rad/s or rev/s

t

dt

d

10.1

• Quick Quizzes p 294

• Angular Acceleration- the rate of change of angular velocity

ort

dt

d

10.1

• Angular Velocity/Acceleration Vector Directions- Right Hand Rule

• Generally CCW is positive, CW is negative

• Acceleration direction Points the same direction asω, if ω is increasing, antiparallel if ω decreases

10.1

Quick Quiz p. 296

10.2 Rotational Kinematics

• Tracking the increasing and decreasing rotation can be done with the same relationships as increasing and decreasing linear motion.

• RememberΔx Δθv ωa α

f i t 21

2f i it t

2 2 2 ( )f i f i

1

2f i i f t

10.2

• Quick Quiz p. 297• Example 10.1

10.3 Angular and Linear Quantities

• When an object rotates on any axis, every particle in that object travels in a circle of constant radius (distance from axis)

• The motion of each point can be described linearly about the circular path

• Tangential Velocity-

dt

dr

dt

dsv

rv

10.3

• Tangential Acceleration-

• We also know there is a centripetal acceleration

dt

dr

dt

dvat

rat

r

vac

2

2rac

10.3

• The resultant acceleration-

• Quick Quizzes p. 298• Examples 10.2

42422222 rrraaa rt

10.4

• Rotational Kinetic Energy- the kinetic energy of a single particle in a rotating object is…

• The Total Kinetic Energy would be the sum of all Ki

• Which can be rewritten...

22212

21 rmvmK iiii

2221 ii

ii

iR rmKK

2221 ii

iR rmK

10.4

• This is a new term we will call Moment of Inertia

• Moment of Inertia has Dimensions ML2 and units kg.m2

(~ rotational counterpart to mass)• Rotational Kinetic Energy-

2ii

irmI

221 IKR

10.4

• Quick Quiz p. 301• Examples 10.3, 10.4

10.5 Calculating Moments of Inertia

• We can evaluate the moment of inertia of an extended object by adding up the M.o.I. for an infinite number of small particles.

dmrmrI iiimi

22

0lim

10.5

• Its generally easier to calculate based on the volume of elements rather than mass so using

for small elements….

• We have…

• If ρ is constant, the integral can be completed based on the geometric shape of the object.

V

m

dVdm

dVrI 2

10.5

• Volumetric Density- ρ (mass per unit volume)• Surface Mass Density- σ (mass per unit area)– (Sheet of uniform thickness (t) σ = ρt)

• Linear Mass Density- λ (mass per unit length)– (Rod of uniform cross sectional area (A)

λ = M/L = ρA)

See Board Diagrams

10.5

• Example 10.5-10.7

• Common M.o.I. for high symmetry shapes (p. 304)

• Parallel Axis Theorem

2MDII CM

10.5

• Example 10.8

10.6 Torque

• Torque- the tendency of a force to cause rotation about an axis

• Where r is the distance from the axis of rotation and Fsinφ is the perpendicular component of the force

• Where F is the force and d is the “moment arm.”

sinFr

dF

10.6

• Moment Arm- (lever arm) the perpendicular distance from the axis of rotation to the “line of action”

10.6

• Torque is a vector has dimensions ML2T-2 which are the same as work, units will also be N.m

• Even though they have the same dimensions and units, they are two very different concepts.

• Work is a scalar product of two vectors• Torque is a vector product of two vectors

10.6

• The direction of the torque vector follows the right hand rule for rotation, and CCW torques will be considered positive, CW torques negative.

• Quick Quizzes p. 307• Example 10.9

10.7 Torque and Angular Acceleration

• Consider a tangential and radial force on a particle.

• The Ft causes a tangential

acceleration.

tt maF

10.7

• We can also look at the torque caused by the tangential force.

• And since…

rmarF tt )(

rat

)( 2mr

I

10.7

• Newton’s 2nd Law (Rotational Analog)

• Quick Quiz p. 309• Review Examples 10.10-10.13

I

10.8 Work, Power, and Energy

• Rotational Analogs for Work, Power, and Energy– Work

– Energy

– Work-KE Theorem

– Power

W

221 IKr

2212

21

if IIW

dt

dW

t

KP r

10.8

• Quick Quiz p. 314• Examples 10.14, 10.15

10.9 Rolling Motion

• For an object rolling in a straight line path the translational motion of its center of mass can be related to its angular displacment, velocity and acceleration.

• Condition for Pure Rolling Motion- no slipping– If there is no slip, then every point on the outside

of the wheel contacts the ground and following relationships hold.

10.9

10.9

10.9

• Linear distance traveled (translational displacment)-

• CofM Velocity (trans. vel.)-

• CofM Accel. (trans. accel.)-

Rs

R

dt

dR

dt

dsvCM

R

dt

dR

dt

dva CMCM

10.9

10.9

• Total Kinetic Energy for a rolling objectKtot = Kr + Kcm

(using just translational speed)

(using just angular speed)

2212

21

CMCM MvIK

2

221

CMCM vMR

IK

2221 MRIK CM

10.9

• Friction must be present to give the torque causing rotation, but does not cause a loss of energy because the point of contact does not slide on the surface.

• With zero friction the object would slide, not roll.

Quick Quizzes p. 319Example 10.16