Lecture 18: Angular Acceleration & Angular Momentum

Questions of Yesterday 1) If an object is rotating at a constant angular speed which statement is true? a) the system is in equilibrium b) the net force on the object is ZERO c) the net torque on the object is ZERO d) all of the above 2) Student 1 (mass = m) sits on the left end of massless seesaw of length L and Student 2 (mass = 2m) sits at the right end. Where must the pivot be placed so the system is in equilibrium? a) L/2 b) L/3 from the right (from Student 2) c) L/3 from the left (from Student 1) d) the system cant be in equilibrium

Torque & Angular Acceleration Force causes linear acceleration TORQUE causes angular acceleration A system is in equilibrium (a = 0,  = 0) when ∑F = 0 and ∑  = 0 What is the induced angular acceleration  of an object due to a certain torque  acting on an object? aT=raT=r  F = ma   = ?   = rFsin 

Torque & Angular Acceleration r FTFT m F T = ma T F T r = mra T aT=raT=r  = r*Fsin   = mr 2  Moment of Inertia (I) of a mass m rotating about an axis at a distance r from the axis = mr 2 mr 2 = Moment of Inertia (I)

Moment of Inertia r2r2 FTFT m1m1 m2m2 r1r1   =  1 +  2 …   = m 1 r 1 2  1 + m 2 r 2 2  2 …  = mr 2    = (m 1 r m 2 r 2 2 …)  = (  mr 2 )   = I  I = m 1 r m 2 r 2 2 … =  mr 2  1 =  2 =  Moment of Inertia depends on axis of rotation!!!

Moment of Inertia of a Rigid Body  = I  mm Mass is distributed over entire body (from 0 to r) I = m 1 r m 2 r 2 2 … =  mr 2 Angular acceleration of every point on rigid body is equal What is the Moment of Inertia of a rotating rigid body?

Moment of Inertia of a Rigid Body What is the Moment of Inertia of a rotating rigid body? m  = I  I =  mr 2 I 1 = m 1 r 1 2 I 2 = m 2 r 2 2   = m 1 r 1 2  1 + m 2 r 2 2  2 …   = (  mr 2 )  Moment of Inertia of a rigid body depends on the MASS of the object AND the DISTRIBUTION of mass about the AXIS of rotation

Moment of Inertia of a Rigid Body Which object has a greater Moment of Inertia? I =  mr 2 r mm r If the same force F is applied to each object as shown… which object will have a greater angular acceleration? FF

Moment of Inertia of a Rigid Body I =  mr 2 In which case is the moment of inertia of the baton greater? mm L axis of rotation mm L If both batons were rotating with the same , and the same braking torque is applied to both… Which one would come to rest sooner?

Rotational Kinetic Energy What is the kinetic energy of a rotating object? r1r1 m I 1 = m 1 r 1 2 I 2 = m 2 r 2 2 r v m KE = (1/2)mv 2 KE = (1/2)m(r  ) 2 KE r = (1/2)I  2 KE r = KE r1 + KE r2 …= ∑KE r KE r = ∑(1/2)mv 2 KE r = ∑(1/2)mr 2  2 KE r = (1/2)(∑mr 2 )  2 KE r = (1/2)I  2

Is energy conserved as the ball rolls down the frictionless ramp? Rotational Kinetic Energy Conservation of Mechanical Energy when W nc = 0 (KE t + KE r + PE G + PE S ) i = (KE t + KE r + PE G + PE S ) f (1/2)mv 2 mgy (1/2)I  2 (1/2)kx 2 What forms of energy does the ball have while rolling down the ramp?

Rotational Kinetic Energy (1/2)mv 2 (1/2)I  2 (KE t + KE r + PE G + PE S ) i = (KE t + KE r + PE G + PE S ) f mgy (1/2)kx 2 Conservation of Mechanical Energy when W nc = 0 Work-Energy Theorem W nc =  KE t +  KE r +  PE G +  PE S

What forms of energy does each object have…. at the top of the ramp (before being released)? halfway down the ramp? at the bottom of the ramp? Rotational Kinetic Energy h h  What is the speed of each object when it reaches the bottom of the frictionless ramp (in terms of m,g, h, R and  )? Which object reaches the bottom first? m m Sphere radius = R I s = (2/5)mR 2 Cube length = 2R

The net TORQUE acting on an object is equal to the CHANGE in ANGULAR MOMENTUM in a given TIME interval Angular Momentum p = mv Linear Momentum = pp tt Relating F & p ∑F∑F ∑F = ma  mv) tt = Newton’s 2nd Law Rotational Analog to Newton’s 2nd Law ∑  = I   I  ) tt = Angular Momentum L = I  Relating  & L = LL tt ∑∑

The net TORQUE acting on an object is equal to the CHANGE in ANGULAR MOMENTUM in a given TIME interval Conservation of Angular Momentum If NO net external TORQUE is acting on an object then ANGULAR MOMENTUM is CONSERVED = LL tt ∑∑ if ∑  = 0 then  L = 0 L i = L f I i  i  = I f  f if ∑  = 0 Conservation of Angular Momentum

You (mass m) are standing at the center of a merry-go-round (I = (1/2)MR 2 ) which is rotating with angular speed  1, as you walk to the outer edge of the merry-go-round… What happens the angular momentum of the system? What happens the angular speed of the merry-go-round? What happens to the rotational kinetic energy of the system? Angular Momentum R M R M 11  2 = ?

A kg cylindrical reel with the radius of m and a frictionless axle starts from rest and speeds up uniformly as a 5.00 kg bucket falls into a well, making a light rope unwind from the reel. The bucket starts from rest and falls for 5.00 s. Practice Problem What is the linear acceleration of the falling bucket? How far does it drop? What is the angular acceleration of the reel? Use energy conservation principles to determine the speed of the spool after the bucket has fallen 5.00 m 5.00 kg 10.0 kg m

Two astronauts, each having a mass of kg, are connected by a 10.0 m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles… What is the magnitude of the angular momentum and the rotational energy of the system? By pulling on the rope, the astronauts shorten the distance between them to 5.00 m… What is the new angular momentum of the system? What are their new angular and linear speeds? What is the new rotational energy of the system? How much work is done by the astronauts in shortening the rope? Practice Problem

Questions of the Day 1) A solid sphere and a hoop of equal radius and mass are both rolled up an incline with the same initial velocity. Which object will travel farthest up the inclined plane? a) the sphere b) the hoop c) they’ll both travel the same distance up the plane d) it depends on the angle of the incline 2) If an acrobat rotates once each second while sailing through the air, and then contracts to reduce her moment of inertia to 1/3 of what is was, how many rotations per second will result? a) once each second b) 3 times each second c) 1/3 times each second d) 9 times each second