Rotational Motion

Rotation of rigid objects- object with definite shape

Rotational Motion

• All points on object move in circles

• Center of these circles is a line=axis of rotation

• What are some examples of rotational motion?

• What is the difference between rotation and revolution?

Speed

• Rotating objects have 2 speeds:

• Linear speed (also known as tangential)

• Rotational speed

Linear Speed

• Imagine yourself on a merry-go-round. At any moment, describe the direction of your linear speed

• Who goes faster- A or B?

Velocity:Linear vs Angular

• Each point on rotating object also has linear velocity and acceleration

• Direction of linear velocity is tangent to circle at that point

• “the hammer throw”

Angular Velocity

• Angular velocity rate of change of angular position

• measured in revolutions/time

Thus RPM= revolutions per minute

Angular Velocity

• All points in rigid object rotate with same angular velocity (move through same angle in same amount of time)

• Related to linear- if you speed up the rotation, both linear and angular velocity increases

Velocity:Linear vs Angular

• Even though angular velocity is same for any point, linear velocity depends on how far away from axis of rotation

• Think of a merry-go-round

So how are they related?

• The farther out you are, the faster your linear speed

• So linear velocity increases with your radius

• The faster your angular speed, the faster your linear speed

• So linear velocity increases with angular velocity

Centripetal Acceleration

• If object is moving in a circle, its direction is constantly changing towards the center so the acceleration must be in that direction

• Then why when you turn a corner in a car do you feel pushed out, not in?

Centripetal Acceleration

• acceleration= change in velocity (speed and direction) in circular motion you are always changing direction- acceleration is towards the axis of rotation

• The farther away you are from the axis of rotation, the greater the centripetal acceleration

• Demo- crack the whip• http://www.glenbrook.k12.il.us/gbssci/phys/

mmedia/circmot/ucm.gif

Centripetal examples

• Wet towel

• Bucket of water

• Beware….inertia is often misinterpreted as a force.

The “f” word• When you turn quickly- say in a car or roller

coaster- you experience that feeling of leaning outward

• You’ve heard it described before as centrifugal force

• Arghh……the “f” word• When you are in circular motion, the force is

inward- towards the axis= centripetal• So why does it feel like you are pushed

out???INERTIA

Centripetal acceleration and force

• Centripetal acceleration– Towards axis of rotation

• Centripetal force– Towards axis of rotation

Frequency

• Frequency= f= revolutions per second (Hz)

• Period=T=time to make one complete revolution

• T= 1/f

Frequency and Period example

• After closing a deal with a client, Kent leans back in his swivel chair and spins around with a frequency of 0.5Hz. What is Kent’s period of spin?

T=1/f=1/0.5Hz=2s

Rolling

Rolling

• Rolling= rotation + translation

• Static friction between rolling object and ground (point of contact is momentarily at rest so static)

Inertia

• Remember our friend, Newton?

• F=ma• In circular motion:

– torque takes the place of force

– Angular acceleration takes the place of acceleration

Rotational Inertia=LAZINESS

• Moment of inertia for a point object

I = Resistance to rotation• I plays the same role for rotational motion as

mass does for translational motion• I depends on distribution of mass with respect

to axis of rotation• When mass is concentrated close to axis

of rotation, I is lower so easier to start and stop rotation

Rotational InertiaUnlike translational motion, distribution of mass

is important in rotational motion.

Rotational inertia- baseball

• A long bat that you hold at the end has a lot of rotational inertia- mass is far away from the axis of rotation

• Thus it is hard to get moving

• Younger players “choke up” on the bat by moving their hands towards the middle- this makes the bat have less rotational inertia- it’s easier to swing

• Try the rotating sticks!

Changing rotational inertia

• When you change your rotational inertia you can drastically change your velocity

• So what about conservation of momentum?

Angular momentum

• Momentum is conserved when no outside forces are acting

• In rotation- this means if no outside torques are acting

• A spinning ice skater pulls in her arms (decreasing her radius of spin) and spins faster yet her momentum is conserved

Torque

How do you make an object start to rotate?

Pick an object in the room and list all the ways you can think of to

make it start rotating.

Torque

• Let’s say we want to spin a can on the table. A force is required.

• One way to start rotation is to wind a string around outer edge of can and then pull.

• Where is the force acting?

• In which direction is the force acting?

Torque

Force acting on outside of can. Where string leaves the can, pulling tangent.

Torque

• Where you apply the force is important.• Think of trying to open a heavy door- if

you push right next to the hinges (axis of rotation) it is very hard to move. If you push far from the hinges it is easier to move.

• Distance from axis of rotation =lever arm or moment arm

Torque

• Which string will open the door the easiest?

• In which direction do you need to pull the string to make the door open easiest?

Torque

Torque• tau = torque (mN)• If force is perpendicular, =rF• If force is not perpendicular, need to find the

perpendicular component of F

=rF

Torque example (perpendicular)

• Ned tightens a bolt in his car engine by exerting 12N of force on his wrench at a distance of 0.40m from the fulcrum. How much torque must he produce to turn the bolt? (force is applied perpendicular to rotation)

Torque= =rF=(12N)(0.4m)=4.8mN

More than one Torque

• When 1 torque acting, angular acceleration is proportional to net torque

• If forces acting to rotate object in same direction net torque=sum of torques

• If forces acting to rotate object in opposite directions net torque=difference of torques

• Counterclockwise +• Clockwise -

Multiple Torque experiment

• Tape a penny to each side of your pencil and then balance pencil on your finger.

• Each penny exerts a torque that is equal to its weight (force of gravity) times the distance r from the balance point on your finger.

• Torques are equal but opposite in direction so net torque=0

• If you placed 2 pennies on one side, where could you place the single penny on the other side to balance the torques?

Torque and center of mass

• Stand with your heels against the wall and try to touch your toes.

• If there is no base of support under your center of mass you will topple over

Center of mass

• The average position of all the mass of an object

• If object is symmetrical- center of mass is at the center of the object

• Where is the center of mass of a meter stick?• A donut?• How could you find the center of mass of an

object?

Torque and football

• If you kick the ball at the center of mass it will not spin

• If you kick the ball above or below the center of mass it will spin