Rotational Mechanics

Rotational Motion

Rotational Speed or Angular SpeedTypically measured in rpm’s or degrees/sec, but the SI unit is radians/sec

ω = Δ Θ / Time

??? What is the rotational speed of a clock’s “second” hand in rpms?

Angular Acceleration:The rate of change in Angular speed

α = Δ ω / Time

Rotational Motion VS Linear MotionThe linear speed or tangential speed of an object

depends on its radial distance. v = ω r

Ex: The horses on a carousel all go around with the same angular speed, but the outside horses have a greater linear

speed.

??? Suppose two cars have a race. Car A has 12 inch wheels that can rotate at 500 rpm. Car B has 24 inch wheels that can rotate at 400 rpm. Which car will win the race?

Measuring Angular Speed1. Record the time it takes an object to roll down

an incline 1 meter long.2. Determine the diameter and circumference of the

object.3. Determine the number of rotations the object

went through when it rolled 1 meter.4. Determine the object’s average Angular Speed.

(rotations/sec)5. Determine the object’s final Angular Speed.

(rotations/sec)6. Determine the object’s angular acceleration.

(rotations/sec2)7. What are some possible reasons why different

objects rolled down at different rates?

Rotation ReviewA wheel starts at rest and then rolls 5

times down a hill in just 3 seconds.

1. What was its average angular speed in rotations/sec?

2. What was its final angular speed in rotations/sec?

3. What was its angular acceleration?

Buttered Cat Paradox

Warm-up

1. Where is the most exciting place to sit on a merry-go-round? Why?

2. If railroad wheels are on the same axle (same angular speed), how is it possible for trains to turn?

3. What should roll down a hill faster, a solid sphere, solid disk or hollow hoop (assume same mass)?

Rotational InertiaIntroduction• Observe the motion of a Ball, Solid Disk, and a

Hoop rolling down an incline.• Rotate the two different “inertia sticks”• Observe the falling chimney.Definition:

Rotational Inertia directly depends on the mass of a rotating object and on how far the mass is located from the axis of rotation.

Other Examples:Tight rope walkers, FlyWheels,Running, Gymnasts

More rotation examples

Ex. If you turn off your workshop grinder and electric drill at the same time, which one takes longer to come to a stop? (2 reasons)

Ex. Why are little kids told to “choke up” on a baseball bat?

Ex. Why do the front brakes wear out faster than the rear brakes on a car?

Torque• Torques cause changes in rotation! Just as

Forces cause accelerations.• Torque = (Force) x (lever arm distance)

– Ex: 100 Newtons of force is applied perpendicularly to a 0.5 meter wrench.

– Remember: the perpendicular component of the applied force is what counts!

– Ex: Torque Feeler, Torque wrench, Cheater Bar?

Torque examplesMeter stick challenge (weighted and

unweighted)HammerSee-sawScrewdriver—how would you modify the

shape to maximize torque?

Center of Mass / Center of GravityCenter of Mass (Gravity) is the average position of all the mass

(weight) that comprises an object (balance point). Ex: Where is the CM of a ball located?

Ex: Where is the CM of a hollow disc located?Ex: Where is the CM of a hammer located?

For freely rotating objects; they will rotate around their CM.Baton twirling, riflesBaby in-uteroEarth-moon system—why does the Earth wobble?

Earth/Moon Sun/Earth

Stability• An object is stable if the sum of the torques

acting on it equal zero (cancel each other out)– Ex: A 2 kg mass is suspended from the 10

cm mark of a meter stick. Where can a 5 kg mass be placed on the meter stick so that it is balanced about the 50 cm mark?

• You can also consider objects to be stable if their CM is located directly above or beneath an area of support.• Ex: Tower of Pisa, Heel/Toes against the

wall, Chair/Head/Wall pick up• Freshmen! Pregnant women!

Stability – Check Questions1. What is the torque caused by a 250 N

force applied at the end of a 40 cm wrench?

2. A 5 kg mass is suspended from the 5 cm mark of a meter stick. Where can a 8 kg mass be placed on the meter stick so that it is balanced about the 50 cm mark?

Centripetal Force and Acceleration• Centripetal Force is a force exerted on an object that

causes it to change direction. It is always directed towards the center of curvature.– ac = v2/r– Fc = mv2/r

– Centrifugal Force is the reaction force.– Ex: A 0.1 kg ball twirled around by a string at a

constant 5 m/s.– What is the centripetal acceleration?– What is the centripetal force on it?– Distinguish the centripetal force from the

centrifugal force.

What On Earth is Spin?

http://www.youtube.com/watch?v=H4JUTnkiGpI&edufilter=8gyTAQpnAaFWN9xxR7pk4A&safe=active

Simulated Gravity

By rotating a space station, inhabitants will be forced to move in a circular path and experience a centripetal force – much like gravity.

However the space station will most likely be large to overcome dizzying effects and varying amounts of Centripetal ForceEx’s: 2001, 2010, Mission Mars, Armageddon

Review

1. What is the weight of the above meter stick shown above?

2. What is the centripetal acceleration of a space station that is 20 meters in radius and rotating at 10 m/s?

10 N

20 cm0 cm 100 cm

Angular Momentum Conservation

• Angular Momentum is Rotational Inertia IN MOTION!– L = I

• Conservation of Angular Momentum:– In the absence of net torque, the total

angular momentum of a system is conserved.

• Ex’s– Spinning stool and tire, Ice Skater,

Gymnast, Earth’s rotation, Comets in orbit. Helicopters…

XKCD fans?

What allows cats to flip over and ALWAYS land on their feet?!?

(video)

Helicopters

Tail Rotor prevents the helicopter from

spinning out of control.

Why don’t these have tail rotors?

Precession• If there is an outside torque acting on a

system, then the angular momentum will change.

• Ex: Hanging bike tire, Earth’s precession (~26,000 yrs), Spinning Top