third physics olympiad 2018 - people.rit.edu · third physics olympiad 2018 march 24, 2018, 12:30...

Third Physics Olympiad 2018

March 24, 2018, 12:30 pm---3:30 pm

Instructions:

You have 3 hours in which to complete this test. Cellular phones may not be used.

Complete only sections A and B if you wish to compete as an Individual. Complete

sections A, B, & C if you wish to compete as a team of one, two, or three (max) students.

You may tear off the formula sheets at the end of the test packet and keep it in front of

you for easy reference. It does not need to be returned with the test packet.

Any calculator (except the one that comes with your cellular phone) may be used.

If more space is needed, ask your teacher for extra sheets. Write your name, team name

(as applicable) and the name of your school on each additional sheet used.

---------------------------------------------------[DO NOT WRITE ON TABLE]----------------------------------------------------

Section A MULTIPLE CHOICE

Section B SHORT ANSWERS

Section C PROBLEM #1



TOTAL SCORE

Test created by Dr. Vivek Narayanan, with contributions from profs. Michael Richmond, Robert Szalapski, and

Gregory Trayling; faculty of School of Physics and Astronomy, RIT.

Your name: ___________________________________________________________

Your team name: ____________________________________________________

Team members: ______________________ ______________________

______________________

School Name: ________________________________________________________

[Blank page: use for rough work; reference problem no. if continuing a problem]

Section A. Multiple choice: [10×3=30] Each question has exactly one correct answer. Clearly circle ONE choice. For this part, your displayed work will not be

graded, only the choice of the correctly circled answer will be graded.

1. A crow flies back and forth at a speed of 60 mph between an express train traveling to the right at an initial speed of

30 mph, and another express train traveling to left on the same track at an initial speed of 30 mph. The trains are 2.0

miles apart when the crow begins this shuttling process, and alerted by this unusual phenomenon, both engineers

realize their danger and apply their brakes at once. If both their trains have the same deceleration, and they come to a

stop just in front of each other, with their locomotive bumpers touching, then the total distance traveled by the crow is

(a) 1.0 mile

(b) 2.0 miles

(c) 4.0 miles

(d) 8.0 miles

(e) It’s infinity, because as the two trains approach, the number of trips made by the crow increases without limit

(f) None of the above answers is correct

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2. An electric bus starts and stops with the same acceleration (or deceleration). If the bus stops are spaced a distance

100 m apart, what is minimum time taken to travel between stops? Neglect any speed limits.

(a) 10.0 seconds

(b) 20.3 seconds

(c) 3.2 seconds

(d) 7.1 seconds

(e) none of the above

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3. Suppose that fully suited up like an astronaut you can jump a height ℎ on Earth. The largest spherical asteroid made of

uniform density material off whose surface you could jump and never fall back is proportional to

(a) ℎ

(b) ℎ2

(c) √ℎ

(d) 1 ℎ⁄

(e) 1√ℎ

⁄

(f) There is not enough information---for example, my mass plus the mass of the space suit needs to be known.

4. A solid sphere and a hollow sphere roll down a semicircular ramp

fixed to the ground at Q. Both spheres are released from rest so

their centers of mass line up with the top left at P. The portion of

the ramp from P to Q is very rough (𝜇 → ∞), while the portion of

the ramp from Q to R is very smooth (𝜇 → 0). Then

(a) Both spheres will arrive at point R.

(b) Neither sphere will arrive at point R, but the solid sphere will come closer to R than the hollow sphere.

(c) Neither sphere will arrive at point R, but the hollow sphere will come closer to R than the solid sphere.

(d) Neither sphere will arrive at point R, but the final heights they attain above the ground Q will be the same.

(e) No such prediction can be made unless we are told how the radii and masses of the spheres compare.

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5. Three identical planets of mass 𝑚 each move in deep space in a stable circular

orbit while maintaining an equilateral triangle configuration. The relationship

between their orbit period 𝑇 and orbit radius 𝑟 is given by

(a) 𝑇2 =4𝜋2

𝐺𝑚𝑟3

(b) 𝑇2 =4𝜋2

3𝐺𝑚𝑟3

(c) 𝑇2 =12𝜋2

𝐺𝑚𝑟3

(d) 𝑇2 =4𝜋2

√3𝐺𝑚𝑟3

(e) 𝑇2 =4√3𝜋2

𝐺𝑚𝑟3

(f) None of the above

P

Q

R

6. An insulating slab of dielectric constant 𝜅 = 4 is inserted into a parallel plate capacitor such that it is parallel to the

plates, and its cross-sectional area is the same as the plates, but it fills only 50% of the vertical space between the

plates. Then which of the following statements is true:

(a) The capacitance of the capacitor will not change.

(b) The capacitance will increase by 50%.

(c) The capacitance will increase by 60%.

(d) The capacitance will increase by a factor of 200%.

(e) The capacitance will actually decrease by 12.5%.

(f) The change of capacitance will depend upon how the slab is positioned vertically between the plates.

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7. The view from above shows an ice hockey puck attached to a flexible string which

attaches in turn to a metal pole partly buried in ice. Practice shots are hit with an

initial velocity 𝑣0. The string wraps around the pole, the puck rebounds, and comes

back to the player for the next practice shot. Assuming the string is never slack, the

speed with which the puck hits the pole is determined by

(a) linear momentum conservation only.

(b) angular momentum conservation only.

(c) energy conservation only.

(d) both energy conservation and angular momentum conservation.

(e) no physics principles; the problem is a purely geometric one.

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8. The cross-and-circle object is made from wire of resistance 1Ω/cm. The radius

of the object is 4 cm. The power delivered by the 6 volt battery is about

(a) 1 W

(b) 5 W

(c) 10 W

(d) 15 W

𝑣0

pole

9. The hoop is held fixed. A string whose length can be varied is tied

between points P and Q on the hoop. A bead of mass 𝑚 is free to

slide without friction along the string. We keep the point P fixed,

but we can vary the angle 𝜃 the string makes with the vertical by

tying it at different points Q. Then the time taken for the bead to

slide from P to Q, assuming the bead does not distort the string,

(a) is minimum when 𝜃 = 0°, that is, when the string is vertical.

(b) is the same for any location of Q, i.e., does not depend on the angle 𝜃.

(c) is the minimum when 𝜃 = 45°.

(d) is an increasing function of 𝜃.

(e) depends on the bead mass 𝑚--- a heavier bead will take less time!

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10.

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[Use this space for rough work]

𝜃

m

P

Q

Section B. Short Answers [5×5=25] Answer each question starting from basic physics concepts, in complete sentences. Use equations as necessary.

1. Using only a meter stick and no timing device of any sort, show how the initial speed of an eraser your teacher knocks

off her horizontal desk may be determined.

2. A metal sheet with a circular hole is heated. What happens to the hole’s diameter? Carefully reason out your answer!

3. A candle in an enclosed bell jar will not burn for long. Describe what would happen instead if the bell jar with the lit

candle inside it is dropped so it falls freely for a long time.

4. A few ice cubes with big air bubbles trapped in them are put in a drink, topping the glass. Predict with reasoning what

will happen to the liquid level when the ice melts---stay the same, go down slightly, or overflow slightly?

5. An inventor creates an alleged perpetual motion device that is

claimed to work as follows: an inverted 3-4-5 triangular wedge

is made perfectly smooth and held fixed. Glass beads strung

together with flexible threading are then draped over the wedge.

Now, since there are more beads on the 4-edge than on the

3-edge, the beads should slide down continuously in a clockwise

fashion, and drag the rest of the bead necklace in the motion, creating unlimited kinetic energy!

Carefully explain the flaw in the inventor’s claims.

<STOP here if taking as an individual. Proceed to the next section only if taking as a team of 1, 2, or 3 student(s).>

Section C. Problems [3×15=45] Select any three of the four problems, clearly crossing out the one not chosen. All details must be shown for full credit. If

you answer all four questions, only the first three will be graded, regardless of how you perform on the fourth. Also, if you

are an individual who attempts this section, then congratulations: you are now a TEAM!

1.

[More space available to work this problem on the next page]

P

It is suspected that a spherical cavity of radius 𝑟 lies buried just beneath the surface

of a uniform density planet of radius 𝑅. To confirm this, you swing a pendulum and

find that the period right above the suspected cavity (point P) is 20% higher than it

would have been had no cavity been present. Determine 𝑟 𝑅⁄ , the ratio of the cavity

radius to the planetary radius. (Your answer will be a number. Box that number.)

[This page is being left intentionally blank, in case you need more space for the first problem]

[Test continues on the following page.]

2.


It is found that when a mass 𝑀 is hung vertically from a certain spring

of unstretched length 𝑙0 = 0.5 m, it stretches by 0.1 m. This same

spring is now used to create the situation shown in the figure. Both

masses are 𝑀, the pulley is ideal, the table is frictionless. The spring is

attached to the ceiling, and is initially unstretched, and nothing is

moving. Someone now comes in with a pair of scissors and cuts the

string to the left of the table mass. Determine the speed, in m/s, of

the table mass when it just lifts off the table.

𝑙0

Cut

[This page is being left intentionally blank, in case you need more space for the second problem]

[Test continues on the following page]

3.


a

b

0

A cube of edge length 𝐿 = 30 cm is fashioned from 3.6 m of copper wire. It is

suspended as shown from a horizontal cable 𝑎𝑏̅̅ ̅, so the fresh paint on it may

dry. A sudden breeze blows and sets the cube oscillating slightly about 𝑎𝑏̅̅ ̅. Find

the length of a simple pendulum with the same oscillation frequency as this

cube.

[This page is being left intentionally blank, in case you need more space for the third problem]

[Test continues on the following page]

4.


The top view and side view of a box-shaped boat are shown. The floor area of the boat

is 𝐴 = 5.0 square meters. With smuggled goods and a couple of smugglers, the loaded

boat has a mass of 500.0 kg. The boat is on very calm waters, when suddenly a Coast

Guard vessel appears. The smugglers dump their valuable cargo of mass 100.0 kg

overboard without a splash. However, the Coast Guard knows physics and is able to

calculate how much cargo was dumped by observing the oscillation of the boat in the

water. What frequency and amplitude of the oscillation did the Coast Guard record?

A

[This page is being left intentionally blank, in case you need more space for the fourth problem]

[STOP: The test has ended. Hand this test to your Instructor. Tear off the formula sheets and keep them with you.]

[Blank page: use for rough work; reference problem no. if continuing a problem]

Formula Sheet for Physics Olympiad (SI units)

§0. Mathematics:

1. If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 then 𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎. 2. A⃗⃗ ∙ B⃗⃗ = AB cos 𝜃 3. ‖A⃗⃗ × B⃗⃗ ‖ = AB sin 𝜃

4. sin2 𝜃 + cos2 𝜃 = 1 5. sin(𝛼 ± 𝛽) = sin𝛼 cos𝛽 ± cos𝛼 sin𝛽 6. cos(𝛼 ± 𝛽) = cos𝛼 cos 𝛽 ∓ sin𝛼 sin𝛽

7. log(𝑚𝑛) = log(𝑚) + log(𝑛) 8. log(𝑚𝑛) = 𝑛 log(𝑚) 9. (1 + 𝑥)𝑛 ≈ 1 + 𝑛𝑥, if |𝑥| ≪ 1

10. sin 𝑥 ≈ 𝑥 ; cos 𝑥 ≈ 1 −𝑥2

2; tan 𝑥 ≈ 𝑥, if |𝑥| ≪ 1 , & 𝑥 is in radians

§1. SI Prefixes: 1. centi (c) 10−2 2. milli (m) 10−3 3. micro (𝜇) 10−6 4. nano (n) 10−9 5. kilo (k) 103 6. mega (M) 106 7. giga (G) 109

§2. SI Units: 1. Length [L]: meter, m 2. Mass [M]: kilogram, kg 3. Time [T]: second, s 4. Temperature [Θ]: kelvin, K

5. Charge [Q]: coulomb, C 6. Magnetic field [B]: tesla, T

§3. Conversions: 1). 1 foot = 12 inches 2). 1 inch = 2.54 cm 3). 1 m/s ≈ 9/4 mph; 1 mph ≈ 4/9 m/s 4). K = °C + 273

5). 1 metric ton = 1000 kg

§4. Kinematics:

1. (velocity) 𝑣 = Δ𝑟

Δ𝑡 2. (acceleration) 𝑎 =

Δ�⃗�

Δ𝑡 3. (centripetal acceleration) 𝑎𝑐 =

𝑣2

𝑅 4. (rotational variables)

𝑠 = 𝑅𝜃; 𝑣 = 𝑅𝜔; 𝑎𝑡 = 𝛼𝑅; 𝑎𝑐 = 𝜔2𝑅. 5. 𝑣 = 𝑣0 + 𝑎𝑡; 𝑥 = 𝑥0 + 𝑣0𝑡 +1

2𝑎𝑡2; 𝑣2 = 𝑣0

2 + 2𝑎(𝑥 − 𝑥0)

§5. Dynamics:

1. (momentum) 𝑝 = 𝑚𝑣 2. (2nd law) 𝐹 = 𝑚𝑎 3. ( kinetic friction) 𝑓𝑘 = 𝜇𝑘𝑁 4. (static friction) 𝑓𝑠 ≤ 𝜇𝑠𝑁

5. (relativistic mass) 𝑚(𝑣) = 𝛾 𝑚(0) 6. (relativistic dilation factor) 𝛾 =1

√1−(𝑣2

𝑐2)

§6. Work and Energy:

1. (work) 𝑊 = 𝐹𝐷 cos 𝜃 2. (power) 𝑃 =Δ𝑊

Δ𝑡 3. (kinetic energy) 𝐾 =

1

2𝑚𝑣2

4. (gravitational potential energy) 𝑈𝑔 = 𝑚𝑔𝑦 5. (spring potential energy) 𝑈𝑠 =1

2𝑘𝑥2

§7. Rotation:

1. (angular momentum) 𝐿 = 𝑚𝑣𝑟 = 𝐼𝜔 2. (torque) 𝜏 = 𝑟𝐹 sin𝜃 3. (2nd law) 𝜏 = 𝐼𝛼 4. (rot. KE) 𝐾𝑟 =1

2𝐼𝜔2

5. (Parallel axis theorem) 𝐼𝑝 = 𝐼𝐶 + 𝑚𝑟2 6. (Common Moments of Inertia about CM): (point mass or ring)

𝐼 = 𝑚𝑅2; (disc) 𝐼 =1

2𝑚𝑅2; (solid sphere) 𝐼 =

2

5𝑚𝑅2; (hollow sphere) 𝐼 =

2

3𝑚𝑅2; (thin rod) 𝐼 =

1

12𝑚𝐿2.

§8. Gravity:

1. (Newton’s law) 𝐹 =𝐺𝑀𝑚

𝑟2 2. (weight) 𝐹 = 𝑚𝑔 3. (grav. PE) 𝑈 = −𝐺𝑀𝑚

𝑟

§9. Elasticity:

1. (pressure or stress) 𝑝 =𝐹

𝐴⫠ 2. (Hooke’s law) 𝐹 = −𝑘𝑥 3. (strain) 𝜖 =

𝛿𝑙

𝑙0 4. (Young’s modulus) 𝑌 =

𝑝

𝜖

§10. Fluids:

1. (density) 𝜌 =𝑀

𝑉 2. (buoyant force) 𝐹𝑏 = 𝜌𝑙𝑔𝑉 3. (Bernoulli’s equation)

1

2𝜌𝑣2 + 𝜌𝑔ℎ + 𝑝 = 𝑐𝑜𝑛𝑠𝑡.

§11. Thermodynamics:

1. (ideal gas law) 𝑝𝑉 = 𝑛𝑅𝑇 2. (first law) Δ𝑄 = Δ𝑈 + 𝑊 3. (mean KE of molecule) 1

2𝑚�̅�2 =

3

2𝑘𝑇

4. (thermal expansion) Δ𝐿 = 𝐿0𝛼Δ𝑇 5. (specific heat) Δ𝑄 = 𝑐𝑚Δ𝑇

§12. Electricity & Magnetism:

1. (Coulomb’s law) 𝐹 =1

4𝜋𝜖0

𝑄𝑞

𝑟2 2. (electric field) �⃗� =𝐹

𝑞 3. (elec. potential) Δ𝑉 = −

𝑊

𝑞 4. (capacitance) 𝐶 =

𝑄

𝑉

5. (cap. energy) 𝑈 =1

2𝐶𝑉2 6. (parallel plate capacitor) 𝐶 = 𝜅𝜖0

𝐴

𝐷 7. (Ohm’s law) 𝑉 = 𝐼𝑅 8. (power) 𝑃 = 𝑉𝐼

9. (resistors in series) 𝑅 = 𝑅1 + 𝑅2 10. (resistors in parallel) 1

𝑅=

1

𝑅1+

1

𝑅2 11. (capacitors in series)

1

𝐶=

1

𝐶1+

1

𝐶2

12. (capacitors in parallel) 𝐶 = 𝐶1 + 𝐶2 13. (resistance of wire) 𝑅 = 𝜌𝑟𝑒𝑠𝐿

𝐴 14. (current) 𝐼 =

Δ𝑄

Δ𝑇

15. (magnetic force) 𝐹 = 𝐼�⃗� × �⃗� 16. (Faraday’s law for induced emf) Ɛ = −ΔΦ𝐵

Δ𝑡 17. (flux) Φ𝐵 = 𝑁𝐵𝐴 cos 𝜃

§13. Oscillations, Waves & optics:

1. (physical pendulum period) 𝑇 = 2𝜋√𝐼

𝑚𝑔𝑑 2. (wave relation) 𝑣 = 𝑓𝜆 3. (Doppler effect for sound)

𝑓𝑠

𝑣±𝑣𝑠=

𝑓𝐿

𝑣±𝑣𝐿

4. (loudness) 𝛽 = 10dB. log𝐼

𝐼0 5. (Snell’s law) 𝑛1 sin𝜃1 = 𝑛2 sin 𝜃2 6. (polarization) 𝐼 = 𝐼0 cos2 𝜃

7. (blackbody radiation) 𝐼 = 𝜎𝑆𝐵𝑇4 8. (thin lens equation) 1

𝑓=

1

𝑑0+

1

𝑑𝑖 ; (magnification) 𝑚 =

ℎ𝑖

ℎ𝑜= −

𝑑𝑖

𝑑𝑜

§14. Useful constants: 1. Electron mass 𝑚𝑒 = 9.1 × 10−31𝑘𝑔 2. Proton mass 𝑚𝑝 = 1.7 × 10−27𝑘𝑔

3. Acceleration due to Earth’s gravity 𝑔 = 9.8 𝑚/𝑠2 4. Avogadro’s number 𝑁𝐴 = 6.0 × 1023

5. Boltzmann constant 𝑘 = 1.4 × 10−23𝐽/𝐾 6. Speed of light in vacuum 𝑐 = 3.0 × 108𝑚/𝑠

7. Permittivity of free space 𝜖0 = 8.9 × 10−12𝐹/𝑚 8. Gas constant 𝑅 = 8.3𝐽

𝐾.𝑚𝑜𝑙

9. Earth’s atmospheric pressure 𝑃𝑎𝑡𝑚 = 1.0 × 105𝑃𝑎 = 14.7 𝑝𝑠𝑖

10. Earth’s mass 𝑀𝐸 = 6.0 × 1024𝑘𝑔 11. Earth’s mean radius 𝑅𝐸 = 6.4 × 106𝑚

12. Newton’s gravitational constant 𝐺 = 6.7 × 10−11𝑁𝑚2/𝑘𝑔2 13. Stefan-Boltzmann constant 𝜎𝑆𝐵 = 5.7 × 108 𝑊

𝑚2.𝐾

14. Density of water, STP, 𝜌𝑊 = 1.0 × 103 𝑘𝑔/𝑚3 15. Density of air, STP, 𝜌𝑎 = 1.23 𝑘𝑔/𝑚3

Brief list of useful atomic masses (in atomic mass units; 1 𝑎𝑚𝑢 ≈ 𝑚𝑝)

Hydrogen H ≡ 1 Carbon C ≡ 12 Nitrogen N ≡ 14 Oxygen O ≡ 16

third physics olympiad 2018 - people.rit.edu · third physics olympiad 2018 march 24, 2018, 12:30...

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