Rotation RETEACH

Main Topics to be Covered

Circular Motion  Remember, uniform circular motion- constant velocity Conical Pendulum A ball of mass m is suspended by a string of length L. The ball revolves with a constant speed v in a horizontal circle of radius r. Determine the horizontal component of tension on the string. Vertical component of T. What is the total amount of tension on the string?

Tarzan Problem Tarzan (m = 85.0 kg) tries to cross a river by swinging from a vine. The vine is 10.5 m long, and his speed at the bottom of the swing (as he just clears the water) is 9.00 m/s. Tarzan doesn't know that the vine has a breaking strength of 1000 N. Find the tension on the vine. Does Tarzan make is across the river?

Riding a Ferris Wheel A child of mass m rides a Ferris wheel. The child moves in a vertical circle of radius 10 m at a constant speed of 3 m/s.  Determine the force exerted by the seat on the child at the bottom of the ride in terms of mg.  Determine the force exerted by the seat on the child at the top of the ride in terms of mg

Rotational Kinematic Equations (all for constant alpha) For any motion given as a function of time

Angular Acceleration The angular speed of an automobile engine is increased from 1200 rev/min to 3000 rev/min in 12s. What is the acceleration in rev/min^2, assuming it to be uniform?

Position Function

Spinning Baseball A good pitcher can throw a baseball at 85 mph with a spin of 1800 rev/min. How many revolutions does the baseball make on its way to home plate if it is 60ft away (and the trajectory is a straight line)?

Unwinding String A pulley wheel 8.0cm in diameter has a 5.6 m long cord wrapped around its periphery. Starting from rest, the wheel is given a constant angular acceleration of 1.5 rad/s^2.  Through what angle must the wheel for the cord to unwind?  How long does it take?

Net Acceleration A car starts from rest and moves around a circular track of radius 30.0m. Its speed increases at the constant rate of m/s^2. What is the magnitude if its net linear acceleration 15 s later?

Rotational Kinetic Energy

Flywheel

Rolling Up a Ramp A solid 8kg sphere rolls up an incline with an inclination of 30 degrees. At the bottom of the incline the center of mass of the sphere has a translational speed of 9.3 m/s.  What is the kinetic energy of the sphere at the bottom of the incline?  How far does the sphere travel up the incline?  If it was a 10kg ball would the height be: less than, greater, or equal to the previous answer?

Rolling with a Loop A small solid marble of mass m and radius r rolls without slipping along the loop the loop track, having been released from rest. The radius of the loop-the-loop track is R with R >>> r  From what minimum height h above the bottom of the track must the marble be released in order that it not leave the track at the top of the loop?

Torque Basics C B D A EF  Torque is produced No torque is produced

Direction of Torque r F θ θ Right-Hand Rule: Shows direction of torque  cross-product r Fsinθ  magnitude + Counter clockwise - clockwise

Net Torque

Difficulty Tightening a Bolt C B A

So what about pulleys? In AP 1 you used pulley systems… BUT ignored mass and size of pulley ( essentially ignoring moment of inertia of pulley itself)

Mass-less Pulley (unrealistic system) A gram mass ( m 1 ) and 50.0-gram mass ( m 2 ) are connected by a string. The string is stretched over a pulley.  Determine the acceleration of the masses and the tension in the string.

Another Mass-less pulley Consider the two-body situation at the right. A 20.0-gram hanging mass (m 2 ) is attached to a gram air track glider (m 1 ).  Determine the acceleration of the system and the tension in the string.

Why are these solutions not correct & impossible??

So what about pulleys? We’ve ignored mass and size of pulley ( moment of inertia of pulley itself) BUT… THE PULLEY IS ROTATING Why is this a crucial piece of information in the system?

Net Torque

Analyzing Tension ALONG Pulleys  If the tension is the same for both sides of the string, what is the Net Torque? T T rr If torque is the tendency to rotate, we know the pulley is rotating, but we calculate 0 N m net torque, is this feasible?

Newton’s Second Law of Rotation

The original solution is wrong because we ignored this rotating pulley. Analyzing Tension ALONG Pulleys T r T T T

Realistic Pulley System A gram mass ( m 1 ) and 50.0-gram mass ( m 2 ) are connected by a string. The string is stretched over a pulley. The pulley is 75 grams and 10cm in diameter.  Determine the acceleration of the masses and the tension in each side of the string.

Another Realistic Pulley System Consider the two-body situation at the right. A 20.0-gram hanging mass (m 2 ) is attached to a gram air track glider (m 1 ).  Determine the acceleration of the system and the tension in the string.

Rolling Down a Ramp A uniform sphere rolls down an incline.  What must be the incline angle if the linear acceleration of the center of the sphere is 0.10g?  For this angle, what would be the acceleration of a frictionless block sliding down the incline?