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2019-2020 BPHO英国物理奥赛第一轮第二部分真题及解析

Category: 国际竞赛, 物理国际竞赛 Date: 2021年11月4日下午6:56

BPhO Round 1

Section 2

15th November 2019

This question paper must not be taken out of the exam room

Instructions Time: 5 minutes reading time (NO writing) and then 1 hour 20 minutes for writing. Questions: Only two questions out of the fifive questions in Section 2 should be attempted. Each question contains independent parts so that later parts should be attempted even if earlier parts are incomplete. Working: Working, calculations, explanations and diagrams, properly laid out, must be shown for full credit. The fifinal answer alone is not suffificient. Writing must be clear. If derivations are required, they must be mathematically supported, with any approximations stated and justifified.

Marks: Students are recommended to spend about 40 minutes on each question. Each question in Section 2 is out of 25, with a maximum of 50 marks from two questions only. Instructions: You are allowed any standard exam board data/formula sheet. Calculators: Any standard calculator may be used, but calculators cannot be programmable and must not have symbolic algebra capability. Solutions: Answers and calculations are to be written on loose paper or in examination booklets. Graph paper and formula sheets should also be made available. Students should ensure that their name and their school/college are clearly written on each and every answer sheet. Number the pages.

Setting the paper: There are two options for sitting BPhO Round 1: a. Section 1 and Section 2 may be sat in one session of 2 hours 40 minutes plus 5 minutes reading time (for Section 2 only). Section 1 should be collected in after 1 hour 20 minutes and then Section 2 given out. b. Section 1 and Section 2 may be sat in two sessions on separate occasions, with 1 hour 20 minutes plus 5 minutes reading time allocated for Section 2. If the paper is taken in two sessions on separate occasions, Section 1 must be collected in after the fifirst session and Section 2 handed out at the beginning of the second session.

Section 2

Question 2 This is about forces on dam walls and detecting cracks in them through wave reflflection. a) The dam shown in 1 is used to generate hydroelectric power. The difference in height between the water levels of the dam is 61 m, and the peak flflow rate of the water through the dam is 3380 m3 s−1 . Estimate the maximum power output of the hydroelectric facility.

credit: https://upload.wikimedia.org/wikipedia/commons/b/b2/Epa-archives the dalles dam-cropped.jpg

b) In practice not all of the gravitational potential energy of the water behind the dam is transferred by the generators, and the water emerges from the dam with a velocity of 8.0 ms ^-1 . Calculate the effificiency of the energy transfer process from the water.

c) The dam can be considered to be a concrete prism of triangular cross-section, with height h = 60 m, a base of thickness t = 60 m, and a width of w = 2700 m. The cross-section of the dam is illustrated in Figure 2. Points F and C are at the bottom of the dam. The water level is at the top of the dam wall.
(i) Calculate the resultant force on the face of the dam due to water pressure in terms of h, w, ρ and g, where ρ is the density of water.
(ii) Calculate the resultant moment (torque) about F due to this force, and calculate the effective height at which the resultant force acts to produce this moment.

The centre of mass of a uniform triangle is at the point of intersection of its medians (the lines joining its vertices to the midpoints of the opposite sides). Thus the centre of mass lies two-thirds of the way along each median.
(iii) For the dam, state the position of the centre of mass of the dam relative to F.
(iv) Calculate the two turning moments acting on the dam about C, due to the water pressure and the weight of the dam.
(v) The frictional force of the dam on the ground beneath is given by f = µN where N is the normal reaction force of the ground, and µ = 0.75. Calculate the minimum density of concrete that could be used so that the dam will not slide over the ground.
(vi) Calculate the minimum density of concrete used to construct the dam so that it will not tip up about point C.
Density of water is ρw = 1000 kgm ^-3
d) There is a signifificant danger of cracks forming in dams, leading to catastrophic failure with little or no warning. One form of non-destructive testing uses ultrasonic waves to investigate the interior of the concrete.
When a wave strikes a boundary (interface) between two different media it has to obey two constraints.

the displacement of the material by the wave must be the same either side of the boundary.

the energy transferred into the boundary must be the same as the total energy reflflected and transmitted.

If an incident wave has an amplitude Ai , the reflflected wave amplitude Ar and the transmitted wave amplitude At , then we can write: The energy of a wave is proportional to the square of the amplitude, A2 , and the rate of energy flflow in a medium is given by ，where Z stands for the acoustic impedance and is a parameter of the medium, and ω is a constant (it cancels in all our equations and can be ignored).
(i) From the energy flflow through a boundary from medium 1 to medium 2, show that where Z1 and Z2 are the impedances of the media either side of the boundary, as illustrated in Figure 3.
Using these two requirements (eq. (1) and (2)) for a wave crossing an interface between two media at normal incidence:
(ii) show that , and
(iii) derive an expression for in terms of Z1 and Z2.
(iv) If Z1 = Z2, what can be said about the transmitted and reflflected waves and the values of and
(v) When Z2 > Z1, what effect does this produce?
The energy reflflected back can be used to charaterise cracks in the concrete. The coeffificient of reflflection, R, is the ratio of the reflflected to the incident energy flflow, such
that R =. The coeffificient of transmission, T, is the ratio of the transmitted energy flflow in medium 2 to the incident energy flflow in medium 1.
(v) Obtain expressions for R and T in terms of Z1 and Z2.
(vi) Calculate a value for R + T.
For a medium, Z = ρvsound, where ρ is the density of the medium, and vsound is the speed of sound in the medium. Values for the densities and sound speeds in air, concrete and water are given in Table 1.
(vii) Calculate the percentage of ultrasound energy reflflected from a crack in the concrete when
i. the crack is fifilled with air,
ii. the crack is fifilled with water.
Question 3

Several mechanical fairground rides are analysed in this question. a) A particle of mass m moves at a constant rate in a horizontal circular path of radius r.State why there must be a resultant force acting on the particle; indicate its direction on a diagram, and state its magnitude both in terms of its speed v and its angular speed ω b) A fairground ride consists of a set of chairs hanging from cables of length ` = 4.0 m, which are attached to circular support of radius a = 6.0 m, as shown in Figure 4 below. Each passenger sits in a light chair, with combined mass m, which can be treated as a point mass at the end of each cable. The ride rotates at angular speed ω, and the cable
hangs at an angle of θ = 30◦ to the vertical. Obtain an expression for the angular velocity ω in terms of `, a, θ and g and then evaluate ω for the given values. c) The “wall of death” ride consists of a d = 10 m diameter drum which rotates at an angular rate ω, shown in Figure 5a. Passengers stand against the inside vertical, rough wall and as the ride speeds up to rate ω, they are held against the wall and cannot slide down due to friction with the wall.

(i) Sketch a diagram of the three forces acting on a passenger who is held against the wall.

(ii) If the frictional force is given by f = µN where N is the normal reaction force of the wall, and µ = 0.4, calculate the minimum value of ω for the drum in order for passengers to be held against the wall.

(iii) The rotating drum is now tilted through 60◦ so that its axis of rotation is θ = 30◦ above the horizontal (Figure 5b). Sketch a diagram of the three forces acting on a passenger when they are at the highest point of the circular path.

(iv) The frictional force f remains as µN with µ = 0.4 although the value of f will be different as the wall is no longer vertical. Given this condition on f, calculate the new value of the minimum angular velocity, ωt of the drum so that passengers are held against the wall at the highest point.

d) Roller coasters operate using gravity and are not powered. A roller coaster similar to that illustrated in

Fig. 6 can be modelled as a circular loop of radius of 20 m connected to a straight sloping track, shown in Figure 7.

Figure 6:

credit: https://www.themeparkreview.com/parks/p 294 2875 gold reef city golden loop

(i) Now use the simplifified model of

Figure 7.

Estimate the minimum height h of the top of the slope from which a car must start in order to remain in contact with the track at all times when the car travels around the loop of radius r = 20 m without falling. Friction can be neglected. Hint: consider the forces acting on the car at the top of the loop, and the minimum speed needed to remain in contact.

(ii) In this simplifified model of the loop ride, the straight track joins on to the circular track at the bottom of the slope. Friction now acts on the carriage such that the frictional force is f = µN, where N is the normal contact force of the track on the carriage and µ = 0.05.
i. Draw a force diagram for the resultant force acting on the carriage as it accelerates down the straight track.
ii. For a track length ` = 60 m at an angle θ = 40◦ above the horizontal, by what percentage does friction reduce the speed of the carriage at the bottom of the sloped track?

(iii) We again neglect friction. The straight track now joins a curved track with a radius of curvature of `/2. What would be the change in the normal contact force acting on a 60 kg passenger from his seat at the moment the carriage enters the curve?

e) A student has an even more unusual idea for a fun-fair ride. This can be modelled as two bars (A and B) each of mass M separated by two springs of negligible mass, each of spring constant k and unstretched length ` . The springs are lightly damped to stop vibrations. This is similar to the example shown in

Figure 8.

The lower bar A is initially at a height of h above the ground. The ride is then released from rest and falls freely under gravity.

Figure 8:

credit: https://www.kerncountyfair.com/p/things-to-do/carnival

(i) Find an expression for the speed of the bars immediately before they hit the ground.

(ii) Bar A strikes the ground and is brought to rest almost instantaneously, its kinetic energy all being transferred to heat and sound. Determine the total energy of the system, E, in terms of the subsequent maximum compression of the springs xm, and gravitational potential energy. After this, the springs extend again, and then at some moment bar A lifts off the ground.

(iii) By considering the forces acting on A determine the extension xe, of the springs at the instant A leaves the ground.
(iv) Hence, using energy considerations, deduce an expression for the condition the initial height h must satisfy in order for A to leave the ground.

Question 4

This question is about electric fifields and their application.
a) A thin, isolated, conducting, square metal sheet of area A is given a charge +Q.

(i) State where the charge on the sheet is to be found.

(ii) Sketch a set of electric fifield lines (with arrows) emerging from the sheet in the region near the surface and centre of the sheet (i.e. neglect edge effects).

(iii) How are the fifield lines qualitatively related to the magnitude of the electric fifield in a given region?

(iv) If the fifield strength near the sheet is Es , write down the force on a charge +q at a distance d from the sheet, where d 《 A, in terms of Es and q.

Consider a conducting spherical shell of radius r given a charge +Q.

(i) i. What is the charge density, σ (the charge per unit area), and the electric fifield strength E at the surface of the sphere?
ii. Hence write down an expression for the electric fifield strength E in terms of σ and ε0 for the conducting, charged surface. How does E depend upon r?

iii. Hence, for an isolated, thin, plane, conducting surface carrying charge density σ, what is the electric fifield strength very near the surface?

(ii) A conducting, charged sphere has no fifield inside the surface for a charge Q on the surface, but a fifield Esphere on the outside surface. The repulsion of the surface charges will cause an outward force on the surface, which can be expressed as a pressure P. Imagine that this sphere described has a small area δA removed, along with its associated charge δq, determined by the value of σ for the sphere, as illustrated in

Figure 9.

The fifield lines remain distributed as before since we are not in practice removing the small area.

i. What is the value of the fifield strength Es on each side of the area δA, in terms of Esphere?

ii. Hence write down the fifield strength in the hole at the spherical surface, just inside where δA would sit, due to the charge on the rest of the sphere, in terms of Esphere.

iii. Hence what is the force on the charge of the area δA when it is in place?
iv. Now derive an expression for the surface pressure on the sphere due to the charge it carries in terms of Esphere and ε0.

An electric fifield strength of greater than 3 × 10⁶ NC ^-1causes air to breakdown and conduct. Determine

(i) the radius of the smallest water drop (treated as a conducting spherical shell so that any excess charge resides on the surface) that can be charged to an electric potential of 7000 V,

(ii) the charge, and the surface charge density of the drop. Charged drops of water are produced in a manner similar to that illustrated in

Figure 10

with the capillary tube at −7.0 kV. The drop is released vertically and falls under gravity towards an earthed metal plate
10 cm below the tube where it was formed.
Calculate
(iii) the speed of the drop when it reaches the earthed metal plate below,

(iv) the average current given a flflow rate through the tube of 64 cm3 per hour.
d)

The intention is to study the motion of the falling drops by means of high speed photography, and one means of doing this is to use a rotating prism camera shutter.

Figure 11

shows the general arrangement for the fifilm passing through the camera at high speed. The continually moving fifilm would blur the image, so a rotating cube of

Figure 11:

credit: (An Overview of High Speed Photographic Imaging by Andrew Davidhazy, School of Photographic Arts and Sciences, Rochester Institute of Technology)

glass is used to move the image for a short time along the fifilm at the same speed as the fifilm travels through the camera.
Figure 12

shows a ray of light entering the rotating cube of glass of side x, with a refractive index n.

(i) Determine an expression for the lateral displacement d, in terms of x, n, and the angle of incidence i, which should be assumed to be small.

(ii) The prism rotates about an axis perpendicular to the plane of the diagram, such that the image is stationary relative to the moving fifilm. How fast should the fifilm move if x = 5.0 cm, n = 1.5 and the cube rotates at 200 Hz? Question 5

This question is on gravity and forces.
a) In 2017 an object that became known as Oumuamua was observed in the Solar System.

Figure 13

shows an artist’s impression. The speed of the object was measured to be 87.7 km s ^-1 at a closest approach to the Sun of 38.1 × 10 ⁶km and astronomers deduced that Oumuamua did not belong to the Solar System. Make a suitable calculation to confifirm this deduction.

b) In January 2019 the New Horizons spacecraft made a flfly by of a distant object in the Solar System called Ultima Thule, shown in

Figure 14

below. The planetoid consists of two lobes of diameters 2r1 = 19.5 km and 2r2 = 14.2 km. The period of rotation about its short axis is T = 15 hours. (The long axis is a line passing through the centres of the two lobes, and a short axis is perpendicular to the long axis.)

If we assume that the two lobes are held together by a weak gravitational attraction, we can model the object as two spheres of the same density just touching.

(i) Find the centre of mass of Ultima Thule, as measured from the centre of the largest sphere.

(ii) By considering the resultant force acting on one sphere, estimate the density of Ultima Thule.

(iii) Using the fact that the density can be taken to be similar for the two lobes, sketch a graph of the gravitational fifield strength along the long axis.
Note: the fifield strength varies linearly with radius inside a spherical body of uniform density.
You may fifind it helpful to sketch the potential along the long axis.

In fact, the two lobes are not bound by gravity, so the density and the masses of the lobes are not known. From the appearance of the two lobes, it looks as though they were moving at a low relative speed when they collided.

As an example, consider two spherical iron masses with the same radii as the lobes of Ultima Thule. The masses are m1 = 3.06 × 1016 kg and m2 = 1.18 × 1016 kg for the larger and smaller lobes respectively. We can assume that they approached from a great distance apart along a line through their centres, with a negligible initial relative velocity.

(iv) Calculate the amount of energy liberated when these two iron spheres collided.

(v) Calculate the relative speed of approach.

This energy of collision may have initially led to an increase in temperature of the two spheres (you may neglect any energy of rotation), which we could calculate from knowing the specifific heat capacity of iron. However, at the very low temperatures of the outer Solar System the molar heat capacity of a material is not constant. For example, at low temperatures the molar heat capacity of iron, c, varies as
where:

c is the molar heat capacity in J mol ^-1 K ^-1
R is the universal gas constant, R = 8.31 J mol ^-1 K ^-1
θD is a constant, θD = 464 K for iron
T is the temperature in kelvin
molar mass of iron = 56 g mol ^-1

(vi) Calculate the number of moles of iron involved in the collision.

(vii) Using your value of the energy liberated on collision, calculate the maximum temperature reached by the iron spheres, assuming the initial temperature was 4 K and radiative losses are in the fifirst instance negligible.
Question 6

This question explores different applications of wave interference.

State the necessary conditions to be able to observe stable wave interference patterns.

(i) Consider a ship that carries a radio receiver a height h above the sea, which is anchored (stationary) a large distance D away from the bottom of a cliff. Suppose on top of the cliff there is a radio transmitter at a height of H above the sea and transmitting waves of wavelength λ. Assuming that the sea acts as a flflat, perfectly reflflecting surface, show that the signal strength received by the ship is a maximum when State any assumptions you make. Note that there is a phase change of π (180◦) when waves reflflect at an interface going from a less dense to a denser medium.

(ii) At high tide, the ship is 1.5 km away from the cliff and its receiver is at 20 m above the sea with the height of the transmitter above the sea at 80 m. The frequency of the radio waves is 70 MHz.
i. By how much would the tide need to fall for the signal received by the ship to become a minimum?

ii. The normal maximum tidal range (twice the amplitude) is 5 m. Would the signal strength change appreciably between low and high tides, and why?

iii. What if the receiver height was just above sea level at 0.5 m?

c) A simplifified model of sodium chloride (salt) consists of a cubic arrangement of alternating sodium and chloride ions separated by distance d. When a plane wave of X-rays is incident on the ions, each ion can be considered as a new source of spherically scattered new waves.

(i) Show that for plane waves of wavelength λ and incident at angle θ to be reflflected from the crystal constructively they must satisfy

2d sin θ = nλ,

where n is a positive integer and θ is specifified as the angle given in

Figure 15.

This is called the Bragg condition.

(ii) The density of salt is 2.17 g cm−3 with a molar mass of 58.4 g mol−1 . Calculate the value of d for salt. (iii) In an X-ray diffraction experiment, waves that are incident at a minimum angle of 25◦ are strongly reflflected from a salt crystal. Calculate the wavelength of the Xrays.

d) An oil fifilm on water produces colourful interference effects, as shown in

Figure 16.

(i) Sketch a diagram showing the path of a ray of light entering and leaving the oil fifilm. Mark on two paths that are going to account for the interference effect observed.

(ii) Calculate the minimum thickness of fifilm needed to observe a strong reflflection of violet light with a wavelength of 410 nm, at an angle of reflflection of 50◦. The refractive index of oil is 1.5, and that of water is 1.33.