06 AL/Structural Question/P.1
HONG KONG ADVANCED LEVEL EXAMINATION
AL PHYSICS
2006 Structural Question
1. Figure 1.1 shows a simplified model of a hydro-electric power station. Water from a reservoir is directed to drive a turbine which is at 12 m below the water level of the reservoir. The turbine rotates at a uniform rate of 7.85 rads-1 and drives a generator through a system shown in Figure 1.2.
(a) (i) The flow rate of water in the uniform pipe is 15 kgs-1. Estimate the power input to the turbine if 90% of the change in gravitational potential energy of water goes to the turbine. (2 marks)
(ii) What is the power output of the turbine if the torque it produces at its axle is 160 Nm? Hence, estimate the efficiency of the turbine in transferring mechanical power. (4 marks)
The motion of the turbine is transmitted to the generator through the system shown in Figure 1.2. A large wheel X with a radius of 0.80 m is connected to a small wheel Y with a radius of 0.02 m through a belt under tension. The wheel X and the turbine have a common axle while the wheel Y and the generator also have a common axle. Assume that there is no slipping between the wheels and the belt.
(b) (i) Find the angular velocity of the axle of the generator. (2 marks)
(ii) The tension in the belt at A is 50 N. Use the information provided in (a)(ii) to find the tension at B. (2 marks)
2. The table below shows the gravitational potential V of the earth at various distances r from the earth’s centre. (Given: radius of the earth = 6.4 ´ 106 m)
6.4 / -62.53
10 / -40.02
20 / -20.01
30 / -13.34
40 / -10.01
(a) (i) Explain the physical meaning of the negative values of the gravitational potential. (1 mark)
(ii) Estimate the gravitational field strength at r = 25 ´ 106 m. (2 marks)
(b) A rocket carrying a spacecraft is launched from the earth’s surface at point A. The spacecraft subsequently enters a circular orbit at a distance of 30 ´ 106 m from the earth’s centre at point B.
(i) Find the orbital speed of the spacecraft. (2 marks)
(ii) The mass of the spacecraft is 5.0 ´ 104 kg. Calculate the change in mechanical energy of the spacecraft between points A and B. (Neglect air resistance and the earth’s rotation.) (3 marks)
(iii) The speed of the spacecraft in this orbit is increased to v’ such that it can leave the earth permanently. Calculate the minimum value of v’.
(2 marks)
(iv) In reality, there is resistance to the motion of the orbiting spacecraft and it gradually falls to a circular orbit with a smaller radius. Express the mechanical energy of the spacecraft in terms of its mass and speed, and explain why its speed still increases despite the resistance. (3 marks)
3. In Figure 3.1, a 50 W slide-wire AB acts as a potential divider. The slide-wire is connected in series with a 2 V d.c. supply of negligible internal resistance and a switch S1. A cell of e.m.f. E and internal resistance r (enclosed by the dotted line), a resistance box R and a switch S2 are connected in series. The cell is also connected to the slide-wire AB through a galvanometer and a sliding contact C.
The procedure to measure the internal resistance r of the cell is as follows:
(i) Set the value of R to 100 W in the resistance box.
(ii) Close the switches S1 and S2.
(iii) Adjust the sliding contact C along the slide-wire AB to obtain zero galvanometer reading.
(iv) Measure the length L of AC on the slide-wire.
(v) Repeat steps (i) to (iv) for different values of R up to 500 W.
A graph of is plotted against using the result obtained.
(a) State the assumption about the slide-wire so that the p.d. across AC is directly proportional to L. (1 mark)
(b) A student suggested that the length L measures the e.m.f. of the cell. Do you agree? Explain briefly. (1 mark)
(c) When the galvanometer reads zero, the current through R is I. Explain why IR is directly proportional to L and hence show that = .
(3 marks)
(d) Use the graph to determine the internal resistance r of the cell. (2 marks)
(e) If a slide-wire of 100 W is used instead, explain the change, if any, on the graph obtained. (2 marks)
4. In a vacuum, a beam of electrons with a horizontal velocity 3.7 ´ 107 ms-1 enters midway into a region of electric field between two horizontal square metal plates as shown in Figure 4.1. The length of the side of the plates is 10 cm. A p.d. of 320 V is applied across the plates and the separation between them is 1.6 cm.
(Given: mass of electron = 9.11 ´ 10-31 kg, electronic charge = –1.60 ´ 10-19 C; permittivity in vacuum = 8.85 ´ 10-12 Fm-l)
(a) Find the electric field strength between the plates. (2 marks)
(b) The electron beam reaches one of the plates at a distance d from the plate’s left edge.
(i) Find the distance d and sketch in Figure 4.1 the path of the electron beam between the plates. (Neglect the weight of the electron.) (5 marks)
(ii) Deduce the corresponding distance d if the applied p.d. is doubled.
(2 marks)
(c) Calculate the charge on the upper plate. (2 marks)
(d) A uniform magnetic field normal to the paper is applied between the plates to make the electron beam travel horizontally. Find the flux density of the magnetic field applied. (Neglect the weight of the electron.) (2 marks)
5. In Figure 5.1, some helium gas is contained in a cylinder fitted with a piston which can move smoothly. The gas has a volume of 300 cm3 at an atmospheric pressure of 1.00 ´ 105 Pa and a temperature of 27°C. The piston is at a height of 20 cm from the bottom of the cylinder. Assume that the gas behaves like an ideal gas.
(Given: universal gas constant = 8.31 Jmol-1K-1; Avogadro constant = 6.02 ´ 1023 mol-1; molar mass of helium = 4.00 g mol-1)
(a) (i) Find the number of moles of helium gas in the cylinder. (2 marks)
(ii) Calculate the r.m.s. speed of the helium molecules. (2 marks)
(b) The cylinder is now placed in a water bath of 90°C.
(i) The cylinder is allowed to reach thermal equilibrium while keeping the pressure of the gas at 1.00 ´ 105 Pa. Find the equilibrium position of the piston, i.e. its height from the bottom of the cylinder. (2 marks)
(ii) If the piston is pushed slowly back to the original position, calculate the new pressure of the gas. (2 marks)
(iii) Sketch a p-V graph to show the sequence of processes in (b)(i) and (b)(ii). Referring to the first law of thermodynamics in the form of DU = Q + W, deduce whether there is heat transfer to or from the gas in the process in (b)(ii). (4 marks)
(iv) If a ring of mass m is placed on the piston so as to maintain the equilibrium in (b)(ii), find m. (3 marks)
6. A light horizontal platform is supported by a light spring fixed vertically on the ground. A coin of mass 0.01 kg is released from rest on the platform. The force constant of the spring is 5 Nm-l. The downward direction is taken to be positive and the coin’s downward displacement from its initial position is denoted by x, as shown in Figure 6.1. (Neglect air resistance.)
(a) Find the equilibrium position and maximum displacement of the coin.
(4 marks)
(b) Sketch a graph of the resultant force F acting on the coin against its displacement x. (2 marks)
(c) Write down the equation of motion of the coin. Explain what kind of motion the coin performs. (3 marks)
(d) At what position does the coin experience the maximum supporting force? Find this maximum supporting force. (3 marks)
(e) Suppose a second coin is placed gently on the first one at the moment when the system is at its highest position during oscillation, state and explain the change in the amplitude and frequency of oscillation. (4 marks)
7. A student uses a grating spectrometer to measure the diffraction angles corresponding to the emission lines from a hydrogen discharge tube. A schematic diagram of the set-up is shown in Figure 7.1. All the necessary adjustments to the spectrometer have already been done.
The angular position of the telescope can be read on the circular vernier scale of the spectrometer table. The smallest division on the main scale is 0.5° while the vernier is marked off in 30 divisions each of which corresponds to ° (see Figure 7.2).
(a) With the grating removed, complete the ray paths from the collimator to the eye. (2 marks)
(b) (i) The diffraction angle q corresponding to each emission line is obtained from the two angular position readings of the telescope taken on either side of the central position. One angular position reading of the first-order red line is 224°30’ and the other is shown on the vernier scale in Figure 7.2. Obtain the other angular position reading and hence find the diffraction angle for this line as well as its percentage error. (3 marks)
(ii) Explain the advantage of taking readings on both sides instead of using just one reading measured from the central position. (1 mark)
(c) The experimental data corresponding to the four visible lines are tabulated below. l denotes the literature values of the wavelengths of those emission lines.
Red / Answer from (b)(i) / 656.3
Cyan / 17.94° / 486.1
Blue / 15.96° / 434.1
Violet / 15.07° / 410.2
(i) Plot a suitable graph to find the grating spacing in nm. (5 marks)
(ii) Find the highest order of diffraction that could be obtained for the violet line. (2 marks)
(d) Figure 7.3 shows some of the allowed energy levels for hydrogen.
Use an arrow to indicate the electron transition that gives rise to the violet line. Find the energy value of the energy level from which the transition starts.
(3 marks)
(Given: Planck constant = 6.63 ´ 10-34 Js, electronic charge = -1.60 ´ 10-19 C)
8. In Figure 8.1, a 47 mF capacitor, an inductor L and a 1 W resistor are connected with a cell of e.m.f. 3 V and negligible internal resistance. The inductor L is of inductance 54 mH and resistance 0.5 W. Initially the capacitor is uncharged.
(a) Find the current flowing in the 1 W resistor
(i) when the switch S is just closed;
(ii) a few minutes after the switch S is closed.
Explain briefly. (4 marks)
(b) (i) Calculate the maximum p.d. across the capacitor. (2 marks)
(ii) Find the energy stored in the inductor at the steady state. (2 marks)
(iii) If switch S is now opened, sketch the time variation of the p.d. VC across the capacitor. (2 marks)
(c) State how you would modify the circuit so as to demonstrate that a large induced e.m.f. is produced across the inductor when switch S is suddenly opened. Explain briefly. (3 marks)
9. Figure 9.1 shows an operational amplifier (op amp). Its typical voltage characteristic showing how the output VO varies with the input (V2 – V1) is also given.
(a) (i) Find the open-loop voltage gain A0 of the op amp. (2 marks)
(ii) The circuit in Figure 9.2 can provide a visual warning when the temperature rises above a predetermined level.
(I) What is component X? Explain how the circuit works. (3 marks)
(II) State how to set a lower predetermined temperature level. (1 mark)
(b) With the following resistors given: 1 kW, 2.2 kW, 4.7 kW, 10 kW, 33 kW, 100 kW, a student tries to investigate the closed-loop voltage gain of the amplifier shown in Figure 9.3. Assume that the op amp is ideal.
(i) Name this type of amplifier and state the function of the combination of resistors R1 and R2. (2 marks)
(ii) Derive the gain of this amplifier in terms of R1 and R2. (2 marks)
(iii) If the gain of the amplifier is to be as near to 20 as possible, which two resistors should be chosen? Calculate the expected gain in this case.
(3 marks)
R1 = ______R2 = ______
(c) Although the closed-loop voltage gain is much smaller than the open-loop one, it is more suitable for most purposes as an amplifier. Give TWO reasons to explain this. (2 marks)
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