Physics Problems Workbook, Exercises of Physics

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Physics Problems

Workbook 2

To help with preparation for the

Physics Aptitude Test (PAT) at the

University of Oxford

Introduction

This is the second workbook full of challenging physics problems designed to help you prepare for the Oxford Physics Aptitude Test (PAT). We hope that you found Workbook 1 (along with its corresponding solutions manual) useful, and that you have some more fun solving the physics problems contained within these pages.

Workbook 2 specifically targets two topics which students often find difficult: circular motion and waves/optics. As before, this workbook contains many ques- tions of varying difficulty and subject matter, and the solutions manual outlines possible approaches to each question in detail. At the end of each chapter in this workbook you will also find hints to help you get started on each question.

The material in this workbook has been written by Dr Justin Palfreyman and Ms Rachel Martin, both A level teachers in UK schools. The workbook and so- lutions manual have been typeset and scrutinised by Lokesh Jain, a Physics and Philosophy undergraduate. Many thanks to them for the hard work involved in producing this material.

Best of luck if you do go on to take the PAT!

Dr Kathryn Boast Access Officer, Physics Department, University of Oxford

Chapter 1

Warm Up Problems

1.1 Warm Up Problems

You will not need a calculator for any of these problems.

1. It takes 20 minutes to fill a bath tub by running the hot water tap. It takes 15 minutes to fill the same bath tub by running the cold water tap. It takes 10 minutes to drain the bath tub by removing the plug. If both taps are running and the plug is removed, how long will it take to fill the bath tub?
2. Find the height of the triangle drawn in Figure 1.1 if a = 9 cm, b = 16 cm and c = 25 cm. Note: the triangle is not drawn to scale!

Figure 1.1: A triangle with lengths a = 9 cm, b = 16 cm and c = 25 cm.

3. A family exercise their dog in the following way:

ˆ The parents sit on a bench while the children walk to a rock a distance D away, calling to the dog as they walk. ˆ The parents and the children alternately call the dog, who runs from one group to the other and back again. ˆ The children then walk back to their parents, still calling the dog on their way back.

If the dog travels at a constant speed vd and the children at a constant speed vc (with vc < vd), calculate how far the dog runs in total.

4. There are 3 cards hidden under a cloth on a table. It is known that one card is white on both sides, one is black on both sides and the other is black on one side and white on the other. I select a card at random and its upper face is white. What are the odds that its other side is also white?
5. Imagine that you have a height h and are standing at a distance d from a mirror, looking at your own reflection. In order to be able to see a full-length view of yourself, the minimum size of the plane mirror must be:

(a) h/ 4 (b) h/ 2 (c) 3h/ 4 (d) h (e) Depends on the exact value of d

6. Zoe wishes to ‘spear’ a fish with a laser – that is, she wants to shine a laser onto a lake and have the light beam hit a fish below the surface of the water. Should she aim the laser beam above, below, or directly at the observed fish to make a direct hit?

Chapter 2

Circular Motion

2.1 Introductory Problems

1. The faster I swing a pendulum around my head, the closer the string gets to being perfectly horizontal.

(a) With the aid of a clear diagram, explain why this is the case. (b) How fast must the pendulum mass be travelling for the string to be exactly horizontal?

2. Assume that the Earth is a perfect sphere of radius 6400 km, spinning on its axis. When a person stands on some weighing scales at the North Pole, the scales read 800 N. We will now think about what would happen if the person were to stand on the scales at different points around the globe.

(a) First qualitatively: if the person were to weigh themselves at the equator, would the reading on the scales be higher, lower, or the same value? (b) Now quantitatively: calculate the difference in the reading on the scales if the person were to weigh themselves at the equator compared to the reading at the North Pole. (c) What would the reading on the scales be if the person were to weigh themselves in Oxford, which has a latitude of 51. 8 ◦^ North?

3. The Earth is actually an oblate spheroid – that is, its equatorial diameter is larger than its North-to-South diameter.

(a) How would this affect the person’s weight at the equator and at the poles? (b) Suggest why the Earth is this shape.

2.2 Further Problems

4. Determine the length of a day in which a person standing on the equator would appear weightless.
5. Newton’s cannon is a thought-experiment whereby a cannonball is fired horizontally from a high mountain top at varying speeds. If the cannonball is fired at or above some critical velocity v, the surface of the Earth will curve away faster than the ball falls back to Earth – the cannonball would now be in orbit.

(a) Determine the orbital velocity. You may assume its orbital radius is 6400 km and ignore air resistance. (b) Hence, or otherwise, determine the period of the orbit.

6. A penny dropped from the top of the Burj Khalifa (height 828 m) in Dubai (latitude 25◦^ North) will miss a target directly below it. Why? By what distance will the penny miss the target?

2.3 Extension Problems

7. A smooth marble is initially at rest at the top of a much larger smooth hemisphere of radius r. The marble is given a slight nudge and begins to slide down the hemisphere.

(a) At what angle from the vertical will the marble leave the surface of the hemisphere? (b) How far away from the base will the marble land?

8. Consider a toy car going around a loop-the-loop (see Figure 2.1). If the car is going too slowly around the loop-the-loop, at some point it will fall off.

(a) If the car started at rest on a downwards ramp which was initially at the same height as the loop, would the car make it around safely? Explain why. (b) Bob wants to find out if it’s possible to do the loop-the-loop with a real car. He has built a loop which has a 6 m radius, and his car will approach the loop driving along a flat runway. What is the minimum speed, in mph, that Bob needs to drive at in order to perform the loop-the-loop successfully? (c) Is there any reason why Bob shouldn’t go much faster than this minimum speed?

9. A velodrome allows cyclists to travel at high speed around tight corners since the track is banked at a steep angle.

13. Imagine that a tunnel is constructed straight through the centre of the Earth. If a person were to fall into the tunnel, would they arrive at the other end? Describe the motion of the person and either explain why the person would not reach the other end or calculate the time taken for the person to travel from one end of the Earth to the other.
14. In The A-Team film, Hannibal and his team find themselves plummeting towards the Earth in a tank with only one of its three parachutes attached. This would not be a soft landing! However, there is a lake about half a mile away from their landing spot. The team attempt to ‘fly the tank’ to the lake by firing shells horizontally. This question will examine whether this is pure Hollywood or based in sound physics. For the team to be successful, how high up must they be when the execute this plan? You may ignore the effects of air resistance in the horizontal direction. The following data may be useful:

ˆ Projectile mass: 10 kg ˆ Muzzle velocity: 1750 ms−^1 ˆ Time between shots: 3.5 s ˆ Tank mass: 22 000 kg ˆ Terminal velocity: 33 mph

2.4 Hints to Circular Motion Problems

2.4.1 Introductory Problems

1. Draw a free body diagram to show the forces acting on the mass. There are only two, but you also know the direction of the resultant force.
2. (a) Recall that scales work by measuring the contact force that they provide on the person standing on them. Draw a free body diagram for the person. Consider whether or not the person is accelerating in each scenario and use Newton’s laws. (b) Remember that the centripetal force is the resultant force – it is always provided by something else. (c) The equator is at a latitude of 0◦^ and the North Pole is at a latitude of 90◦. Have you drawn a cross-section of the Earth and included a right-angled triangle? In which direction does your weight and contact force act? Do you need an additional force? What would happen if you were on an ice-rink?
3. (a) It is helpful to know Newton’s law of gravitation:

F =

GM m r^2

How does the gravitational field strength vary with distance from the centre of mass? (b) Have you ever seen a chef making a pizza base? Similar physics is involved here.

2.4.2 Further Problems

4. What is the condition on the contact force that leads to the experience of weightlessness? How is the resultant force dependent on the period?
5. If there is only one force acting on an object in a circular orbit, this force must also be the centripetal force.
6. This is a synoptic question – circular motion and...what else is involved? Can you work out the difference in linear speed between the top and bottom of the building?

2.4.3 Extension Problems

7. (a) What do you know about the contact force at the point of interest? (b) Have you resolved your initial velocity into useful components? The marble in a fishbowl question from Workbook 1 may be a useful guide.

Figure 2.2: If the person is at radius r, then the gravitational pull from all the mass of the Earth contained outside of this radius exactly cancels out.

there is some mass above them, which is now pulling them upwards. But this exactly cancels out the pull of the other mass outside radius r. In other words, it is only the mass that is contained within a sphere of radius equal to the person’s displacement from the centre of mass that contributes to the force of gravity. This means that you only have to consider the mass within the person’s radius. Recall that simple harmonic motion occurs if the acceleration of an object is proportional to, and in the opposite direction to, the object’s displace- ment about the equilibrium position.

14. Make as many simplifying assumptions as you can. How important is it that the mass of the tank will decrease? Can you smooth out the force? Instead of having many impulses every 3 .5 s, consider finding an average continuous force. If you can find a constant acceleration then regular SUVAT equations can be applied.

Chapter 3

Waves and Optics

3.1 Introductory Problems

1. A triangular glass prism sits on a table pointing upwards. A beam of coloured light is directed horizontally near the top of the prism, as shown in Figure 3.1. What happens to the light beam at the prism?

(a) It is bent upwards (b) It is bent downwards (c) It continues horizontally (d) It depends on the colour of the light

Figure 3.1: A beam of coloured light directed horizontally towards the top of a triangular glass prism.

2. A beam of light is incident from a vacuum onto a medium at an angle θ to the normal of the boundary. The refracted and partially reflected beams happen to form a right angle. Find an expression for the refractive index of the medium.

5. In a particle physics experiment, light from a particle detector is to be collected and concentrated by reflecting it between a pair of plane mirrors with angle 2α between them, as shown in Figure 3.3. A faint parallel beam of light consisting of rays parallel to the central axis is to be narrowed down by reflection off the mirrors, as shown by the single ray illustrated, for which angle a = α.

Figure 3.3: A parallel beam of light being reflected between a pair of plane mirrors.

(a) Determine angles b, c, d and e in terms of angle α. (b) Explain what happens after several reflections of the light down the mirror funnel. (c) If α = 10◦, what is the total number of reflections between the mirrors that will be made by a beam of light entering parallel to the axis of symmetry as shown? (d) If the mirrors are replaced by an internally silvered circular cone whose cross-section is the same as that shown above, why will this not make any difference to the calculation given above for the plane angled mirrors with a beam of light parallel to the axis? (e) An ear trumpet was a device that was used to collect sound and focus it into the ear. It was a cone about 0.5 m long with an angle 2α of about 30◦. The sound passing into the device would typically have a frequency of 400 Hz and a speed of 330 ms−^1. Why is the model above that we have used for light not valid for an ear trumpet used to collect sound?

3.3 Extension Problems

6. Consider the diagram in Figure 3.4. Indicate clearly the position and nature of the image formed by the mirror. Draw rays corresponding to light coming from the open circle, and mark any relevant angles.
7. A parallel sided slab of medium B and refractive index nB is sandwiched between two slabs of medium A of refractive index nA. A beam of light passes from A through B and into A on the other side. If the beam is

Figure 3.4: Indicate clearly the position and nature of the image formed by the mirror. Draw rays corresponding to light coming from the open circle, and mark any relevant angles.

incident on B at an angle of θ to the normal, what is the angle to the normal of the light beam in A after it has left B?

(a) cos−^1

nA sin θ nB

(b) θ

(c) sin−^1

n^2 A sin θ n^2 B

(d) sin−^1

nA sin θ nB

(e) n nAB θ

8. A parallel beam of monochromatic light, initially travelling in a direction above the horizontal, enters a region of atmosphere in which the refractive index increases steadily with height. Which of the graphs in Figure 3. represents the path of the beam of light?
9. A narrow beam of light is incident normally upon a thin slit. The light that passes through is spread out by diffraction. The thin slit is then

Figure 3.7: Two mirrors X and Y and a solid object with white spots at P and Q.

(b) In which mirror would an observer at A see an image of spot Q? Mark S, the position of this image. (c) An observer at B can see an image of P resulting from reflections at both mirrors. Draw a ray of light from P to B which enables this image to be seen.

11. A fisherman listens to the radio as he sits on the bank waiting for a fish to bite. The sound is also heard by the fish and the path of the sound waves entering the water is shown in Figure 3.8.

Figure 3.8: The path of the sound waves entering the water.

(a) Describe what happens to the frequency, wavelength and speed of sound as it moves from air to water. (b) The fisherman’s radio has two speakers, as shown in Figure 3.9.

Figure 3.9: The speaker of the fisherman’s radio.

Sketch a diagram illustrating how destructive interference between sounds from the two speakers can occur when the radio is playing a note of a single frequency, assuming that the waves from the two speakers start in phase.

12. (a) Intensity decays as one moves further away from a source, due to the rays diverging. If I is the intensity and r is the distance from the source, then I ∝ rn^ for what value of n? (b) Rayleigh scattering is an effect that causes many optical phenomena. It is caused by the scattering of light by small particles, such as molecules that make up the air in the atmosphere. If a beam of intensity I 0 and wavelength λ interacts with one of these particles, then the intensity of the light scattered at an angle θ is proportional to I 0 λmrnα^6 (1 + cos^2 θ) (3.1) where r is the distance from the scattering particle and α is the diameter of the scattering particle. The relationship between the intensity of the scattered light (for a given wavelength) with the distance from the scattering particle is the same as for a point source. By considering the dimensions of the quantities involved, what is m to one significant figure?
13. A glass prism of refractive index n = 1.40 has a triangular cross section with two angles of 45◦. The prism floats on some mercury with its largest side of length l = 45.0 cm facing downwards and a vertical depth of h = 2 .50 cm submerged.

(a) A monochromatic beam of light, entering the glass parallel to the mercury surface, internally reflects off the bottom face of the prism due to the presence of the mercury. What is the maximum height of the incident beam above the mercury surface such that the beam can leave on the other side of the prism, parallel to the mercury surface?