Final Exam During class (1 - 3:55 pm) on 6/27, Mon Room: 412 FMH (classroom)

Bring scientific calculators No smart phone callltculators are allowed . Exam covers everything learned in this course.

Review session: Thursday during class



-- -- Kinetic , , mechanical energy for -- Conservation of mechanical energy for rotation -- Angular -- Conservation of


1 How to find torque

F From figure, θ F T m τ = rFTp== rFsinθ r F r

rp axis Line of

If we know components G G rrr= (0)( x ,,y 0) FFF= (0)( x ,,y 0) G τ =−(0,0,rFx yyx rF )

Reminder: Rotational variables are vectors, having direction

The angular , , and ( θ, ω, α ) are vectors with direction.

The directions are given by the right-hand rule:

Fingers of right hand along the angular direction (See Fig.)

Then, the direction of thumb is thdihe direct ion o fhf the angu lar quantity.

2 iClicker quiz A All have the same magnitude. Which has the greatest magniditude of torque wi ihth respect to the pivot? Pivot B

(a) A (b) B C (c) C (d) All forces have the same torque (e) Not enough information

G ‰ 5.1.G A particle located at the vector rmm = (1 , 2 ) has a force F = (2NN ,3 ) acting on it. The torque in N.m about the origin is?

3 Gravitational force on extended objects

Torque on extended object by gravitational force Æ Assume that the total gravitational force effectively acts at the center of .

Gravitational potential energy of extended object Æ M g H, where H is the height of the and M is the total mass.

iClicker Q

Axis of rotation

HiHorizonta l unif orm rod of length L & mass M Find the torque by gravitational force.

A. LMg B. (L/2)Mg C. 2LMg D. (3/2)LMg E. None of the above 1 Find the . I = ML2 end, rod 3

4 Example: Torque and Angular Acceleration of a

•Cord wrapped around disk, hanging • Cord does not slip or stretch •Let m = 1.2 kg, M = 2.5 kg, r =0.2 m • Find acceleration of mass m, r find angular acceleration α for disk, tension, and torque on the a disk 1 Formula: I = Mr2 2 mg

Example: A uniform circular disk of radius r and mass M is pulled by constant horizontal force F applied to the center of mass, and is without slipping. M=2 kg, r=0.5 m, I_(cm,disk)=(1/2)MR^2, F=5N. a) Find the angular acceleration. b) Find the static force.

cm r CCW = + F P


iClicker: If there’s no friction from the surface on the disk, and if the disk is initially not rotating , would it rotate about center of mass by applied force?

A) Yes B) No

5 for rotation

If the rotates with respect to an axis fixed in 1 KI= ω 2 2

Kinetic energy for rotation If the rigid body rotates with respect to a moving axis through the center of mass, such as rolling ω

KK=+rot K cm vcm 1 KI= ω 2 rot2 cm 1 KMv= 2 cm2 cm For rolling without slipping ΔsR=Δθ

vRcm = ω

6 For any rotational or translational motion,

UMgravity= gh cm

Emech =+KUg

Δ=EWmech non− conservative

is conserved. If Wnon− conservative = 0 , then Emech

Example of

Axis of rotation Initial

Final Horizontal uniform rod of length L & mass M is released from rest.

Find its angular speed at the lowest point, assuming no friction between axis of rotation and the rod.

7 Example: Use energy conservation to find the speed of the bowling ball as it rolls w/o slipping to the bottom of the ramp Given: h=2m ƒ Formula: For a solid sphere I = 2 MR2 cm 5 Hint:

ƒ Rotation accelerates if there is friction between the sphere and the ramp ‰ Friction force produces the net torque and angular acceleration. ‰ There is no mechanical energy change because the contact point is always at rest relative to the surface, so no is done against friction

iClicker Q: A solid sphere and a spherical shell of the same radius r and same mass M roll to the bottom of a ramp without slipping from the same height h.

True or false? : “The two have the same speed at the bottom.”

A) True B) False. Shell is faster. C) False. Solid sphere is faster. D) Not enough information.

I_(cm, spherical shell) = (2/3) MR^2 I_(cm, solid sphere)=(2/5) MR^2

8 Angular momentum – concepts & definition

- Linear momentum: p = mv

- Angular (Rotational) momentum: L = of x angular = Iω

linear rotational inertia m I speed v ω rigid body linear p=mv L=Iω angular momentum momentum

Angular momentum of a bowling ball ‰ 6.1. A bowling ball is rotating as shown about its mass center axis. Find it’s angular momentum about that axis, in kg.m2/s

A) 4 B) ½ C) 7 D) 2 E) ¼

ω = 4d/4 rad/s M = 5 kg r = ½ m I = 2/5 MR2 L = Iω

9 Angular momentum of a 2 L== Iωω mr = mvTpTp r = mvrsin θ = mvr = p r = pr

vT = ω r O G r r p vT G θ v

Note: L = 0 if v is parallel to r (radially in or out)

LI==ωθ mvrmvrTp =sin = mvr

= rpTp== rp sinθ r p

Angular momentum of a point particle

O G L = rp== rpsinθ r p r r T p p pT G θ p

If we know components G G ppp= (0)( ,,0) rrr= (0)( xy,,0) x y G L =−(0,0,rpx yyx rp )

10 Angular momentum for car

‰ 5.2. A car of mass 1000 kg moves with a speed of 50 m/s on a circular track of radius 100 m. What is the magnitude of its angular momentum (in kg • m2/s) relative to the center of the race track (point “P”) ?

A) 5.0 × 102 A

6 B) 5.0 × 10 B 4 C) 2.5 × 10 P D) 2.5 × 106 E) 5.0 × 103

‰ 5.3. What would the anggpular momentum about point “P” be if the car leaves the track at “A” and ends up at point “B” with the same velocity ? A) Same as above B) Different from above

C) Not Enough Information L = r ⊥ p = p ⊥r = pr sin( θ)

Net angular momentum of particles

LLLLnet =+++123...

11 Example: calculating angular momentum for particles

PP10602-23*: Two objects are moving as shown in the figure . What is their total angular momentum about point O?



ΔΔω L τα==II = ΔLt=Δτ net net ΔΔtt

Conservation of angular momentum :

Angular momentum, L is consreved if τ net = 0 Conservation of net angular momentum for a system

Net angular momentum, Lnet is consreved if τ net,ext = 0

12 A puck on a frictionless air hockey table has a mass of 5.0 g and is attached to a cord passing through a hole in the surface as in the figure. The puck is revolving at a 2.0 m from the hole with an of 3.0 rad/s. The cord is then pulled from below, shortening the radius to 1 .0 m . The new angular velocity (in rad/s) is ______.

Demonstration: Spinning Professor: Web link

Isolated System

τ net, ext =⇒0 ⇒ L = constant

Iiiω = I fω f

Moment of inertia changes Another link

13 Internal do not change total angular momentum...... G it is redistributed within the isolatedG system τnet = 0 about z - axis ⇒ Lsys = constant Demonstration: Weblink, more Spacecraft maneuvers by Internal torques not 0 spinning the

they reverse L wh the craft count er-rotttates

G G G G G L = 0 = L + L Lboy,final = 2Lwheel tot craft flywheel

Tethered Astronauts

‰ 6.3. Two astronauts each having mass M are connected by a mass-less rope of length d. They are isolated in space, orbiting their center of mass at identical v. By pulling on the rope, one of them shortens the distance between them to d/2. What are the new angular momentum L’ and speed v’?

A) L’ =mvd/2= mvd/2, vv=’=v/2 v/2 B) L’ = mvd, v’ = 2v C) L’ = 2mvd, v’ = v D) L’ = 2mv’d, v’ = v/2 E) L’ = mvd, v’= v/2

L = Iω

L = mv Tr

14 Example: A merry-go-round problem

A 40-kg child running at 4.0 m/s jumps tangentially onto a stationary circular merry-go-round platform whose radius is 2.0 m and whose is 20 kg-m2.

Find the angular velocity of the platform after the child has jumped on.