Rotational MotionRotational Motion&&

TorqueTorque

Angular DisplacementAngular Displacement Angular displacement is a measure of

the angle a line from the center to a point on the outside edge sweeps through as the object rotates

We use the greek letter “theta” to represent angular displacement

Angular displacement is measured in “radians”

Arc LengthArc LengthArc length is

represented by the letter s

it is the distance a point on the edge of the object rotates through

measured in meters

rs

s

r

r

Angular VelocityAngular VelocityAngular velocity is a measure of the

rate of change of the angular position or the “spin rate”

We use the greek letter lowercase “omega” () to represent angular velocity mathematically

Angular velocity is measured in “radians per second” (rad/sec)

Angular VelocityAngular Velocity

r

r

t

May also be expressed in rpm

(revolutions per minute) but must

be converted to rad/sec for calculations

2 1min1

min 1 60sec 30sec

rev rad rad

rev

Angular AccelerationAngular AccelerationAngular acceleration is the rate of

change of angular velocity We use the greek letter “alpha” to

represent angular accelerationAngular acceleration is measured in

radians per second per second (rad/sec2)

Angular AccelerationAngular Acceleration

r

r

f i

t t

Rotational Motion Rotational Motion RelationshipsRelationships

222 if

tif

22

1 tti

2i f t

Conversions and signsConversions and signs• For rotating objects, clockwise (cw) is

negative and counter-clockwise (ccw) is positive. This applies to Ɵ, ɷ, and α.

• Conversions:

180

1 2 2

T

T

s r

v r

a r

rad

rev r rad

What is Torque?What is Torque?

Torque is a measure of how much a force acting on an object causes that object to rotate.

Torque is dependent on force and lever arm and is measured in Newton-meters (Nm)

Lever ArmLever ArmDistance measured

perpendicularly from the line of force to the pivot point.

Measured in metersLever arm

F1 F2pivot

Lever arm1 2

Calculating TorqueCalculating Torque

FlTorque = Force * lever-arm

The symbol for torque is the greek letter “tau”

pivot

F

Note: Force and lever arm must be perpendicular to each other

Calculating TorqueCalculating TorqueBy finding the component of By finding the component of

force perpendicular to dforce perpendicular to d

dFFl )sin(

F

pivot

F

d F

F

F - the perpendicular component of the force

F// - the parallel component of the force – it does not cause torque (lever arm = 0)

Or Calculating Torque Or Calculating Torque by finding the lever armby finding the lever arm

sinFdFl

pivot

d F

=dsin

“d” is the distance from where the force is applied to the pivot point

“” is the angle between d and the line

of F

Net Torque Net Torque

The net torque is the sum of all the individual torques.

Torque that is clockwise (cw) is negative

and torque that is counter-clockwise (ccw)

is positive.

1 2 ...net

Rotational EquilibriumRotational Equilibrium

• In rotational equilibrium, the sum of all the torques is equal to zero. In other words, there is no net torque on the object.

• There is no angular acceleration. • The object is either not rotating or it is

rotating at a constant speed.• or0)( cwccwnet cwccw

( 0)

Linear EquilibriumLinear Equilibrium

• In linear equilibrium, the sum of all the forces is equal to zero. In other words, there is no net force on the object.

• There in no linear acceleration. (a = 0)• The object is either not moving linearly

or it is moving at a constant velocity.• and0netF ,( 0net xF , 0)net yF

Total EquilibriumTotal Equilibrium

• In total equilibrium, both net force and net torque are equal to zero. In other words, there is no net force or net torque on the object.

• and 0net 0netF

EXAMPLE PROBLEM ON EXAMPLE PROBLEM ON TORQUE: The Swinging DoorTORQUE: The Swinging Door

Question In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you exerted on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0 m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

Hints 1. Where is the pivot point? 2. What was the force applied? 3. How far from the pivot point was the force applied? 4. What was the angle between the door and the

direction of force?

Solution The pivot point is at the hinges of the door, opposite to where you were pushing the door. The force you used was 50N, at a distance 1.0m from the pivot point. You hit the door perpendicular to its plane, so the angle between the door and the direction of force was 90 degrees. Since

then the torque on the door was: τ = (1.0m) (50N) sin(90°)

τ = 50 N m

sinFl Fd