Physics 2210 Name:

Fall 2013 Signature:

Final Exam 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. (4 pts) Consider two vectors ~A and ~B as shown in the figure below. Graphically determine

the vector ~C = ~A − 2 ~B. Clearly label the vector ~C in the figure.

2. (3 pts) A box sits on a table of height h above the floor. A person raises the box to a height2h, moves it horizontally, then drops it. Just before the box hits the floor, state whether thequantity is positive, negative or zero.

• (1) Net work done by the person on the box

• (1) Net work done by gravity on the box

• (1) Net work done on the box

3. (3 pts) Consider the wheel in the figure below. It is initially rotating in the counterclockwisedirection, and its angular speed is being decreased by a constant frictional torque. On thediagram, use arrows to indicate the:

(a) direction of the tangential velocity ~vT of point P .

(b) direction of the centripetal acceleration ~ac of point P .

(c) direction of the tangential acceleration ~aT of point P .

4. (4 pts) If global warming continues, it is likely that some ice from the Earth’s polar ice capswill melt and the water will be distributed closer to the equator. If this occurs, would thelength of the day (a) increase (b) decrease or (c) stay the same? Explain.

Student name: UID:

5. (3 pts) Newton’s 3rd Law states that for every action there is an equal and opposite reaction.For a block of mass m sitting on a table, what is the reaction to the gravitational force mg

acting on the block? Explain.

6. (4 pts) A simple pendulum swings through an angle ±θmax such that its motion can beregarded as simple harmonic. State the angle(s) at which each of the following quantities isat a minimum:

(a) (1) Potential energy

(b) (1) Magnitude of angular acceleration

(c) (1) Kinetic energy

(d) (1) Tension in the string

7. (3 pts) An object on the end of a string is executing circular motion, with a tangential speedv. If the length of the string is doubled and the velocity remains unchanged, what wouldhappen to the tension in the string?

8. (4 pts) Two identical guitar strings are vibrating, but string #1 is vibrating with twice theamplitude of string #2. What is the ratio of maximum kinetic energies of the vibratingstrings?

Student name: UID:

9. (8 pts) A projectile is launched from the top of a 218 meter cliff with an initial speed of32.0 m/s at an angle of 25.0◦ above the horizontal. The projectile strikes a wall 210.0 metersaway, measured along the horizontal axis as shown in the figure below.

(a) (4) Calculate the height h at which the projectile strikes the wall

(b) (4) Calculate the magnitude and direction of the projectile’s velocity as it strikes thewall.

Student name: UID:

10. (8 pts) A 455 kg roller coaster starts from rest a height h above the low point on its track.It rolls without friction to the low point, at which its speed has increased to 32.5 me-ters/second. It then travels up a frictionless ramp a height of 14.0 meters, and is slowed to3.0 meters/second by a 1300 N frictional breaking force acting over a distance d. Finallythe car bounces off a massless spring-loaded bumper at the end of the track, with springconstant 1,550 N/m.

(a) (2) Calculate the starting height h of the roller coaster car.

(b) (4) Calculate the work done by the frictional force.

(c) (2) Calculate the maximum compression of the spring.

Student name: UID:

11. (9 pts) A block of mass m = 60.2 kg slides on a surface inclined at 35.0◦ with coefficient ofkinetic friction µk = 0.500, and is attached to the end of a cord which is wrapped around auniform disk of mass M = 186.0 kg and radius R = 0.330 meters. The cord has zero massand it does not stretch, nor does it slip on the disk.

(a) (6) Calculate the acceleration of the block m.

(b) (3) Calculate the magnitude of the centripetal acceleration of a point on the edge ofthe disk, 1.40 seconds after the block begins moving.

Student name: UID:

12. (10 pts) A mass m1 = 4.39 kg on a frictionless horizontal surface is attached to a masslessspring of spring constant 24.5 N/m. Initially, the spring is at equilibrium and the mass m1

is at rest. As second mass m2 = 1.67 kg is incident on this system. m2 collides with m1, andthe two masses stick together, moving to the left with a speed of v0 = 1.40 m/s immediatelyafter the collision.

(a) (3) Calculate the initial speed v2 of mass m2.

(b) (3) Calculate the amplitude of oscillation of the conjoined masses.

(c) (4) Write down an equation for the position of the masses as a function of time. Taket = 0 to be the time of the collision.

Student name: UID:

13. (9 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 205 N.

(a) (6) Calculate the mass of the beam.

(b) (3) Calculate the magnitude of the total force FH exerted by the hinge on the beam.

Student name: UID:

14. (16 pts) A string is attached to the wall at one end, while the other end hangs over a pulleyL = 0.625 meters away as shown in the figure. A mass M hangs from the end of the string.The string has mass per unit length of 0.0120 kg/m, and the speed of a wave on the stringis 620.0 meters/second.

(a) (4) Calculate the value of the mass M .

(b) (4) What is the lowest (fundamental) frequency standing wave that could vibrate onthis string? What are the second and third lowest frequencies?

(c) (4) If the mass M were doubled, what would be the fundamental frequency?

(d) (4) If the string were replaced by a string made of the same material but with half thediameter, what would be the fundamental frequency?

Student name: UID:

15. (12 pts) The simple pendulum shown is released from rest at an angle θ = 14.5◦. It isobserved to swing back to its initial position in 1.30 seconds.

(a) (2) What is the length L of the pendulum?

(b) (4) Write down the equation for the angular displacement of the pendulum as a functionof time.

(c) (6) The largest tension the pendulum string can experience before breaking is 134.0 New-tons. Given that, and the fact that the string doesn’t break, what is the largest possiblevalue of the mass m of the pendulum bob?

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!"