HOKKAIDO UNIVERSITYFACULTY OF SCIENCE

Autumn 2017 Examination PeriodPhysics I [002599] November 21, 2017

1.00p.m. – 2.30p.m.

Student name:

Student ID:

Instructions:You have 90 minutes to attempt to answer all of the questions in this examination. Youare permitted the use of calculators and dictionaries (paper and electronic) but not mobilephones. Communicating with other students, or the use of additional material, is strictlyprohibited and will be considered cheating, resulting in a failing grade.

Part A: contains 8 multiple choice questions [4 marks each, 32 total]Part B: contains 3 long questions [10 marks each, 30 total]

You should answer both parts.

Please show all of your working, as marks will be awarded for sensible and logical attemptsat problem solving, even if the final answer is not correct.

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Part A: Multiple choice

Problem 1. Wheeeeeee!

A student is trying to select which frictionless slide below will give them the greatest speedwhen they reach the bottom. Which one should they chose?

A) Slide 1

B) Slide 2

C) Slide 3

D) Slide 4

E) It doesn’t matter.

[2 marks]

The student finds a different slide as shown below. After sliding down past point Q whathappens to the speed and acceleration in the direction of motion?

A) Both decrease

B) Speed decreases, acceleration increases

C) Both remain constant

D) Speed increases, acceleration decreases

E) Both increase

[2 marks]

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Problem 2. Ball impact

Two steel balls (A and B) collide as shown below, resulting in a change in their motion.Which set of arrows (1–5) best represents the direction of the change in momentum of eachball?

A) 1

B) 2

C) 3

D) 4

E) 5

[2 marks]

Which diagram represents the impulse applied to B by A?

A) 1

B) 2

C) 3

D) 4

E) 5

[2 marks]

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Problem 3. Where did you ichi-go to?

John, Paul and George are standing in a strawberry field. Paul is 14.0m due west of John.George is 36.0m from Paul, in a direction of 37◦ south of east from Paul’s location. What isthe distance between George and John?

A) 38.6m

B) 26.2m

C) 47.9m

D) 29.8m

[3 marks]

What is the direction of George’s location from that of John (measured from south of east)?

A) 56◦

B) 28◦

C) 26◦

D) 15◦

[1 marks]

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Problem 4. Springs and gravity, and friction, oh my!

A 4.0kg block hangs by a light string that passes over a massless, frictionless pulley andis connected to a 6.0kg block that rests on a shelf. The coefficient of kinetic friction forthe shelf–block contact area is 0.2. The 6.0kg block is pushed against a spring with a forceconstant of 180N/m, and is compressed by 30cm. Find the speed of the blocks after the6.0kg block is released and the 4.0kg block has fallen a distance of 40cm.

The block is not attached to the spring, only pushed against it. Assume the 6.0kg blockdoes not collide with the pulley.

A) 3.1m/s

B) 2.4m/s

C) 6.2m/s

D) 2.0m/s

[4 marks]

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Problem 5. Tuning a piano

You are trying to find out the suitability of a piano wire of length 3 m with a linear massdensity of 0.00250 kg/m. You have already found two adjacent resonant frequencies at 252 Hzand at 336 Hz, but we want to know about the fundamental frequency of the wire. Find thefrequency of the fundamental mode. Remember piano wires are fixed at both ends whenthey oscillate!

A) 168Hz

B) 84.0Hz

C) 126Hz

D) 42.3Hz

[3 marks]

Find the minimum tension the wire must be able to withstand to play that fundamentalfrequency.

A) 1429N

B) 2540N

C) 161N

D) 635N

[1 marks]

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Problem 6. The harmonic motion formerly known as simple

We have attached some uniform rod of mass M and length L to a pivot at its centre, andattached each end to springs of spring constant k to fixed surfaces. The rod is free tooscillate about the pivot. If the rod is gently pushed through a small angle, what is theangular frequency of the oscillations, ω?

HINT: You can construct an equation of motion for the rod by considering the two differentdefinitions of torque we considered in the lectures. Then re-express this into the general formof simple harmonic motion.

A)√k/M

B)√

2k/M

C)√

6k/M

D)√

12k/M

[4 marks]

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Problem 7. Snell’s window

You are at the pool and dive under the water. You look up and notice that objects above thewater level are in a circle of light of radius 2.0m, while the rest of your vision is the colourof the sides of the swimming pool. How deep are you below the surface of the water?

HINT: npool = nH2O = 1.33.

A) 1.7m

B) 2.3m

C) 0.6m

D) 3.5m

[4 marks]

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Problem 8. Double lenses

An object is placed 4.0cm to the left of a double convex lens that has a focal length of 12cm.A second lens with a focal length of 6cm is placed 12cm to the right of the first lens. Whatis the position of the final image from the original object as a result of both lenses?

A) 6cm

B) 25cm

C) 33cm

D) 9cm

[3 marks]

Is the final image smaller, larger or the same size as the original object?

A) smaller

B) larger

C) the same size

D) impossible to know

[1 marks]

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Part B: Long questions

Problem 9. A Sunday drive

Sally is driving along a straight highway in a 1965 Mustang. At t = 0, when she is movingat 10m/s in the positive x-direction, she passes a signpost at x = 50 m. Her x-accelerationas a function of time is

ax = 2.0 m/s2 − (0.10 m/s3)t

i) Find an expression for the x-velocity, vx, and x-position, x, as a function of time.

[2 marks]

ii) At what time is vx the greatest (tvmax)? What is the value of vx at this time? Whatdistance is her x position at this time?

[3 marks]

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iii) Draw approximate graphs of ax, vx and x. Indicate the position of tvmax in each plot.

HINT: Drawing the graphs in this order will help!

[3 marks]

iv) At what time will the car come to a stop?

[2 marks]

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Problem 10. The I-scream-cone

A small block with mass m is placed inside an inverted cone that is rotating about a verticalaxis such that the time for one revolution of the cone is T . The walls of the cone make anangle β with the horizontal. The coefficient of static friction between the block and the coneis µs. If the block is to remain at a constant height h above the bottom tip of the cone, thereis some maximum and minimum values of T (i.e. Tmax and Tmin).

i) Draw force diagrams for the block in the cases of Tmax and Tmin.

[2 marks]

ii) Find expressions for Tmax and Tmin as a function of β, h, g (the acceleration due to gravity)and µs. Note: the solution may look quite ugly.

[7 marks]

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iii) What happens to your solutions if we were to instead use a frictionless surface? Brieflyexplain your interpretation of this result.

[1 mark]

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Problem 11. The last dam question

A dam holds back water (of density ρ) at a height of H and has a width of w. We want totry and find out the force of the water behind the dam.

i) If we define height as y with y = 0 as the base of the dam, write an equation that relatesthe pressure, P , to the density of water ρ behind the dam at a given height y (where y ismeasured from the bottom of the dam).

[2 marks]

ii) Using your expression above, derive the total force water exerts on the dam.

[6 marks]

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iii) What is the average pressure on the dam from the force exerted by the water?

[2 marks]

END OF EXAM QUESTIONS

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Useful equations and constants

Name Symbol Value

Gravitational acceleration g 9.81 ms−2

Gravitational constant G 6.67× 10−11 m3kg−1s−2

Speed of light in a vacuum c 3.00× 108 ms−1

Avagadro’s number NA 6.02× 1023 mol−1

Atomic mass unit mu 1.66× 10−27 kg

Table 1: Useful physical constants.

The solutions to quadratic equations of the form: 0 = ax2 + bx+ c are given by:

x =−b±

√b2 − 4ac

2a

The moments of inertia for a selection of objects with mass M and uniform density: