SESSIONAL EXAMINATIONS: 2012W T1

Course Number: PHYS 111 Duration: 3 HOURS

Course name: CALCULUS BASED PHYSICS I

DECEMBER 11, 2012

Instructors: Dr. Bobowski & Dr. Rosolowsky

Candidate Name: __________________________________________________

Student Number: __________________________________________________

Special instructions: Do not take any part of the exam with you when you leave. Work all problems on the examination.

This examination consists of 11 pages (not including this cover sheet). Once the exam begins, check to ensure that it is complete and all pages are printed. There is a formula sheet as the last page of the exam. You may detach the formula sheet but leave the rest of the exam stapled.

RULES GOVERNING FORMAL EXAMINATIONSThe following are the rules governing formal examinations:

1. Each candidate must be prepared to produce, upon request, a UBCcard for identification.

2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.

3. No candidate shall be permitted to enter the examination room after the expiration of one half-hour from the scheduled starting time, or to leave during the first half-hour of the examination.

4. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: • having at the place of writing any books, papers or memoranda, calculators,

computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners;

• speaking or communicating with other candidates; • purposely exposing written papers to the view of other candidates or imaging

devices. The plea of accident or forgetfulness shall not be received.5. Candidates must not destroy or mutilate any examination material, must hand in all

examination papers, and must not take any examination material from the examination room without permission of the invigilator.

6. Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.

Candidate SignatureMark a box indicating your section.

Prof. Rosolowsky • Sec 001 Lecture @ 11:30 a.m.

Prof. Bobowski • Sec 002Lecture @ 9:30 a.m.

PHYSICS 111

2012/13 Term 1Final Examination Name:

Student Number:

Final Exam (80 points)Multiple Choice: Select the best answer for each of the following questions.

Write your answer as an English letter to the left of each problem.

(2pts) 1. A particle has a trajectory given as ~r = [(3t

3 � 2t

2)̂ı� 12 t

4|̂] m for t given in seconds. What is

the magnitude of the acceleration vector for this particle at t = 4.00 s?

(a) 181. m/s2 (b) �42.0 m/s2 (c) 118. m/s2 (d) 54.0 m/s2 (e) 96.0 m/s2

(2pts) 2. Two raindrops in a cloud collide perfectly inelastically. The first raindrop has a mass of 0.40 gand is travelling with ~v1 = (3.00ı̂ � 18.0|̂) m/s. The second raindrop has a mass of 0.30 g andis travelling with ~v2 = (�2.00ı̂ � 14.0|̂) m/s. What is the resulting velocity of the combinedraindrop?

(a) (0.50ı̂ � 16|̂) m/s (b) (0.60ı̂ � 11|̂) m/s(c) (0.14ı̂ � 16|̂) m/s (d) (0.00ı̂ � 11|̂) m/s(e) (0.86ı̂ � 16|̂) m/s

4 pts

PHYS111/FINAL – Page 2 of 11 – Name:

(2pts) 3. Balls A and B are launched along a pair of tracks as shown in the figure. The balls are launchedwith the same initial velocity. Which ball reaches the end of the track first? Neglect friction.

(a) A

(b) B

(c) They reach the end of the track at the same time.

(d) More information is needed.

(2pts) 4. A body of mass 1.00 kg is thrown upwards with a velocity of 20.0 m/s. It momentarily comesto rest after attaining a height of 18.0 m. How much energy is lost due to air friction?

(a) 18.0 J (b) 23.4 J (c) 32.6 J (d) 54.3 J (e) 76.6 J

(2pts) 5. In a lab, a test mass with m = 150±3 g is measured to have a speed of 2.70±0.05 m/s. Whatis the kinetic energy of the mass?

(a) 0.547 ± 0.021 J

(b) 0.547 ± 0.020 J

(c) 0.547 ± 0.052 J

(d) 0.547 ± 0.031 J

(e) 0.547 ± 0.011 J

6 pts

PHYS111/FINAL – Page 3 of 11 – Name:

(2pts) 6. A car of mass m is driving around a horizontal circular track of radius R at constant speed v.The frictional force between the tires of the car and the road is at its maximum value. Whichof the following expressions can be used to find the value of the coe�cient of friction betweenthe tires and the road?

(a) µ = vgR (b) µ =v

2

gR

(c) µ =gv

R

(d) µ =gR

v

(e) µ =mv

2

R

(2pts) 7. Two blocks are arranged as shown and kept at rest by holding the 1 kg block in place. Thetension in the string is closest to which of the following choices?

(a) 9 N (b) 10 N (c) 11 N (d) 12 N (e) 13 N

(2pts) 8. Two identical springs are attached to two identical masses. Both masses are free to slide along africtionless horizontal surface. One of the springs is displaced a distance d from its equilibriumposition, while the other is displaced a distance 2d. If both masses are released at the sameinstant, which of the springs, A or B, returns to its unstretched length first?

(a) Spring A

(b) Spring B

(c) It’s a tie.

(d) Not enough information is given.

6 pts

PHYS111/FINAL – Page 4 of 11 – Name:

(2pts) 9. The potential energy of a spring is given by U(x) = (3x2 � 3x � 6) J if x is given in metres.What is the equilibrium length of the spring?

(a) x = 12 m (b) x = 0 m (c) x = �3

2 m (d) x = �2 m (e) x = 1 m

(2pts) 10. A roller-coaster travels over a vertical, circular loop of radius R = 12.0 m. If it travels withoutfriction along the track, what is the minimum speed required at the bottom of the loop suchthat the roller-coaster remains in contact with the track at the top of the loop?

(a) 10.8 m/s (b) 11.5 m/s (c) 15.3 m/s (d) 18.8 m/s (e) 24.3 m/s

4 pts

PHYS111/FINAL – Page 5 of 11 – Name:

Free Response: Write out complete answers to the following questions. Show your work sinceit allows us to be generous with partial credit. Unless otherwise specificed, significant figure

rules should be used in numeric answers.

(10pts) 11. In the mobile app “Cut the Rope”, a mass (of candy) swings on a rope and the game playerselects a point to cut the rope so it lands in a cute little monster’s mouth. Imagine that themass is suspended from a fixed pivot point by a massless string of length L = 0.200 m. It isreleased from an angle ✓0 = 45.0�, swings through its lowest point, and is then cut on the otherside at ✓

cut

= 30.0�. Once cut, the mass flies free (no drag) and lands on a surface a distanced away from the where it was when the rope was cut. The surface is at the same height as themass when the rope is cut. What is the distance d?

✓0✓cut

L

L

d

10 pts

PHYS111/FINAL – Page 6 of 11 – Name:

(10pts) 12. A wood crate sits in the back of a truck. The coe�cients of friction between the crate and thetruck are µ

s

= 0.900 and µ

k

= 0.500. The truck starts moving up a 20.0� slope. What is themaximum acceleration the truck can have without the crate slipping out the back?

10 pts

PHYS111/FINAL – Page 7 of 11 – Name:

(10pts) 13. A block of mass m1 slides across a frictionless surface with speed v1 and collides perfectlyelastically with a block m2 (initially at rest). Block m2 is attached to a spring with springconstant k and is in equilibrium at x = 0. Assume m2 > m1. Answer the following questionsusing the variables m1, m2, k and/or v1.

(a) (4 points) What is the amplitude of the resulting simple harmonic motion?(b) (2 points) What is the period of the resulting simple harmonic motion?(c) (4 points) What is the position of m2 as a function of time? Note that the +x direction

is defined as to the right in the diagram below. Assume t = 0 at the time of collision.

k

m2m1

v1

x

=0

+x

10 pts

PHYS111/FINAL – Page 8 of 11 – Name:

(10pts) 14. Because of your excellent work in lab, your professor asks you to test out a new “frictionless”air track, which can be used to measure the acceleration due to gravity. In your setup, youplace a cart with mass m1 = 200.0 g on the track and suspend a second mass m2 = 40.00 gover a (supposedly) massless, frictionless pulley. Using a photogate that reads the accelerationof the system at the pulley, you measure the acceleration in 10 trials as given in the tablebelow. The uncertainty in each acceleration measurement is ±0.02 m/s2; the uncertainties inm1 and m2 are negligible. The mean and standard deviation (SD) of the acceleration data arealso given in the table. Given these data, would you conclude that the system is frictionless asclaimed? Justify your answer!

Trial Accel. (m/s2)1 1.652 1.713 1.644 1.665 1.486 1.347 1.708 1.649 1.52

10 1.50Mean 1.58

SD 0.12

10 pts

PHYS111/FINAL – Page 9 of 11 – Name:

(10pts) 15. A 10 g bullet is fired with a speed of 400 m/s toward a solid wood door. The force on thebullet as it passes through the door is shown by the graph below. Find the speed with whichthe bullet exits the opposite side of the door. Significant figures do not count in this problem.

10 pts

PHYS111/FINAL – Page 10 of 11 – Name:

(10pts) 16. The graph below shows the position versus time plotted for a block oscillating on a spring. (a)What is the phase constant �0 for this oscillation? (b) On the first set of empty axes below,draw the velocity versus time graph for the block. Make sure your graph aligns properly withthe position versus time graph above it. (c) On the second set of axes, draw the position versustime graph if the spring constant is decreased by a factor of four but no other conditions arechanged. (d) On the third set of axes, draw the position versus time graph for an oscillation(with the original spring constant) in which the value of �0 is replaced by ��0.

t

x

t

x

(c)

(d)(b)

t

x

t

x

v

10 pts

PHYS111/FINAL – Page 11 of 11 – Name:

Potentially Useful Formulae

You may detach this sheet from the exam.

d

dt

(At

n) = A n t

n�1Z

tf

ti

ct

n =c

n + 1

⇣t

n+1f

� t

n+1i

⌘

~v =d~x

dt

~a =d~v

dt

=d

2~r

dt

2

�x =

Ztf

ti

v dt �v =

Ztf

ti

a dt

~r

f

= ~r

i

+ ~v

i

�t +1

2~a�t

2~v

f

= ~v

i

+ ~a�t

v

2f

= v

2i

+ 2a(xf

� x

i

) x

f

= x

i

+(v

f

+ v

i

)

2�t

~

F

net

=X

~

F = m~a =d~p

dt

~

F

AB

= �~

F

BA

a

r

=v

2

r

= !

2r s = r✓ v

t

= r! a

t

= r↵ ✓

f

= ✓

i

+ !

i

�t +1

2↵�t

2

f

s

µ

s

n f

k

= µ

k

n f

r

= µ

r

n

~

F

s

= �k�~s

~p = m~v

~

J =

Ztf

ti

~

F

net

dt = ~p

f

� ~p

i

For elastic collisions with m2 initially at rest:

v1f =

✓m1 � m2

m1 + m2

◆v1i and v2f =

✓2m1

m1 + m2

◆v1i

W =

Zs2

s1

~

F · d~s =

Zs2

s1

F cos ✓ ds = �K

K =1

2mv

2U

grav

= mgh U

spring

=1

2k(x � x0)

2F

s

= �dU

ds

! = 2⇡f =2⇡

T

x(t) = A cos(!t + �0) + C ! =

rk

m

! =

rg

L

ax

2 + bx + c = 0 has solutions x =�b ±

pb

2 � 4ac

2a

~

A · ~

B = A

x

B

x

+ A

y

B

y

+ A

z

B

z

x̄ =1

n

nX

i=1

x

i

�

n�1 =

vuuutnP

i=1(x̄ � x

i

)2

n � 1�

m

=�

n�1pn

0 pts