Physics 2210 Name:

Fall 2013 Signature:

Exam #3 UID:

Please read the following before continuing:

• Show all work in answering the following questions. Partial credit may be given for problemsinvolving calculations.

• Be sure that your final answer is clearly indicated, for example by drawing a box around it.

• Be sure that your cellphone is turned off.

• Your signature above indicates that you have neither given nor received unauthorized assis-tance on any part of this exam.

• Thanks, and good luck!

Student name: UID:

1. (8 pts) Is it possible for the angular velocity of a (not-necessarily rigid) system to changeeven if there is no net torque on the system? If not, explain why not. If so, explain underwhat circumstances this could occur.

2. (8 pts) Two points, A and B, are marked as shown on a disk rotating with a constant angularacceleration. For each of the following quantities, state whether point A has a greater value,point B has a greater value, or if A and B have the same value at some arbitrary time.

(a) Linear speed

(b) Angular speed

(c) Centripetal acceleration

(d) Tangential acceleration

3. (9 pts) Consider the gyroscope shown in the figure. A disk of mass M rotates upon itsaxis with angular speed ω in the direction shown, a distance d from its point of support.Indicate the direction of the following from your viewing perspective as “left”, “right”, “intothe page” or “out of the page”:

(a) Angular momentum vector

(b) Torque due to gravity

(c) Precession of gyroscope

Student name: UID:

4. (15 pts) A rotating object has three parts. Each part consists of a rod of length L = 2.0 mand mass MR = 12.0 kg, and a hoop of radius R = 0.9 m and mass MH = 3.0 kg, connectedas shown. The object is attached to a fixed base and rotates about its center. Calculate themoment of inertia of the object in units of kg · m2.

Student name: UID:

5. (15 pts) A hollow spherical ball of radius R rolls from rest without slipping down a 32.0◦

ramp of height 1.55 m. Calculate the linear velocity of the hollow ball when it reaches thebottom of the ramp.

Student name: UID:

6. (15 pts) A rotating disk of radius 0.250 meters accelerates uniformly from 33.0 rpm to78.0 rpm. A point on the edge of the disk travels a total of 32.5 meters during this acceler-ation.

(a) (5) Calculate the angular acceleration of the disk

(b) (5) Calculate the centripetal acceleration of a point on the edge of the disk, at t =5.00 seconds after the acceleration starts.

(c) (5) If the mass of the disk is 55.0 kg, calculate the net torque required to achieve thisacceleration.

Student name: UID:

7. (15 pts) A constant force FT = 335 N is applied tangentially at the outer edge of a soliddisk of mass m1 = 50.0 kg and radius R1 = 0.600 meters, accelerating it from rest to a finalangular velocity of 22.4 revolutions per second. The accelerated disk is brought into contactwith a second disk of mass m2 = 4m1 and radius R2 = 2R1 which is initally rotating in theopposite direction of the first disk with an angular velocity of 0.805 revolutions per second.

(a) (7) Calculate the time interval over which the force FT was applied.

(b) (8) Calculate the final angular velocity of the joined system. Specify the magnitudeand direction of rotation.

Student name: UID:

8. (15 pts) A beam, connected to a side wall by a hinge, is held in place with a single wire asshown in the diagram. The tension in the wire is 155 N.

(a) (8) Calculate the mass of the beam.

(b) (7) Calculate the x- and y-components of the force FH exerted by the hinge on thebeam.

Kinematics

g = 9.81m

s2= 32.2 ft

s2

!v =!v0+!at

!x =!x0+!v0t + 1

2

!at2

v2= v

0

2+ 2a x ! x

0( )

!

"##

$##

!a const.

!vA,B

=!vA,C

+!vC ,B

Uniform Circular Motion 2 2v

ra r

v r

!

!

= =

=

Dynamics !Fnet

= m!a =

d!p

dt!FA,B

= !

!FB,A

F = mg (near Earth’s surface)

Fgravity = 1 2

2

m mG

r(in general)

(where G = 6.67 x 10-11

m3 kg

-1 s

-2)

Fspring = -kx

Friction !! ! !!! (kinetic)

!!!!!!! (static)

Work & Kinetic Energy

W =

!F id

!l!

W =

!F i!!r = F!rcos!

(constant force)

( )2 21

2 12

grav

spring

W mg y

W k x x

= ! "

= ! !

K = 1

2mv

2=p2

2m

WNET

= !K

Potential Energy

21

2

grav

grav

spring

NC

(near Earth)

(general)

U mgy

MmU G

r

U kx

E K U W

=

= !

=

" = " + " =

System of Particles

!R

CM=

mi

!rii

!mii

!!V

CM=

mi

!vii

!mii

!!F

ext= M

total

!A

CM!Ksystem,lab

= Krelative to CM

+ KCM

Momentum !Ptotal

= Mtotal

!VCM

d!Ptotal

dt=

!Fnet,external

!Fnetdt! = "

!p =!Favg"t

If !F

net,external= 0, then

!P

total is

constant

Elastic collisions 21

2system i iiK mv=! is conserved

!v2 f!!v1 f

=!v2i!!v1i

!v1i

*=!v1 f

* , !v2i

*=!v2 f

*

Rotational kinematics , , s R v R a R! " #= = =

! =!0+!

0t + 1

2!t

2

! =!0+!t

!2=!

0

2+ 2! " !"

0( )

"

#$$

%$$

! const.

Rotational Dynamics 2 2, parallel CM

i iiI m r I I MD= = +!

2

hoopI MR= Idisk

= 1

2MR

2

21

12

21

3

rod-CM

rod-end

I ML

I ML

=

=

22

5

22

3

solid-sphere

hollow-sphere

I MR

I MR

=

=

!! =!r !

!F

!! = I

!"

21

2

21

2

rotation

translation CM

K I

K MV

!=

=

Statics !F! = 0,

!!! = 0 (any axis)

Angular Momentum !L =!r !!p

!L = I

total

!!

!Ltotal

=

!LCM

+

!L*

d!Ltotal

dt=!!net

If !!

net, external= 0 then

!Ltotal

is

constant 2

21

2

2

LK I

I!= =

externalprec

L

!" =

Simple Harmonic Motion

( )

( )

( )2

( ) cos

( ) sin

( ) cos

x t A t

v t A t

a t A t

! "

! ! "

! ! "

= +

= # +

= # +

(mass-spring)

(simple pend.)

(physical pend.)

(torsion pend.)

km

g

L

mgR

I

I!

"

"

"

"

=

=

=

=

General Harmonic

Transverse Waves

( )( , ) cosy x t A kx t!= "

2k !

"=

22

Pf!" != =

k Pv f ! ""= = =

Waves on a String 2 2

2 2 2

1d d

dx d

y y

v t=

wave

Tv

µ=

2 2

maxK A!"