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PJC P2 1 (a) Define velocity and acceleration. [2] (b) Fig. 1.1 shows the variation with time t of the velocity v for an object.

Fig 1.1 (i) State the time at which the object is at maximum displacement from the starting point. [1] 6.0 s (ii) Calculate the displacement of the object at t = 12.0 s. [1] – 20 m (iii) Sketch a graph to show the variation with time t of the displacement s for the object. (You are not expected to label values of the displacement.) [2]

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2 A solid iron sphere of density 8000 kg m–3 and volume 4.50 x 10–4 m3 is completely submerged in a liquid of density 800 kg m–3. The iron sphere is resting on a spring, as shown in Fig. 2.1. The spring is compressed by 10.2 cm.

Fig. 2.1

(a) Show that the upthrust on the iron sphere is 3.53 N. [1] (b) Hence, calculate the spring constant of the spring. [2] 312 N m–1

(c) A string of breaking strength 32.0 N is used to lift the iron sphere vertically upwards, as shown in Fig. 2.2. The iron sphere is then lifted partially out of the liquid as shown in Fig. 2.3.

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(i) Explain why the string breaks. [1] (ii) Calculate the volume of the fluid displaced at the instant when the string breaks. [2] 4.23 x 10–4 m3 3 (a) A wine glass can be shattered through resonance by maintaining a certain frequency of high intensity sound wave. Fig. 3.1 shows the side view of a wine glass vibrating in response to such a sound wave. On Fig. 3.2, sketch a possible standing wave pattern on the rim of the glass as seen from the top. [2]

(b) The speed v of a progressive wave is given by the expression v = fλ. A stationary wave does not have a speed. By reference to the formation of a stationary wave, explain the significance of the product fλ for a stationary wave. [3] (c) Explain what is meant by diffraction of a wave. [2]

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(d) A narrow beam of coherent light of wavelength 589 nm is incident normally on a diffraction grating having 4.00 ×102 lines for every 1 mm. (i) Determine the number of orders of diffracted light that are visible on each side of the zero order. [2] 4 (ii) A student suspects that there are in fact two wavelengths of light in the incident beam, one at 589.0 nm and the other at 589.6 nm. 1. State the order of diffracted light at which the two wavelengths are most likely to be distinguished. [1] 2. The minimum angular separation of the diffracted light for which two wavelengths may be distinguished is 0.10°. By means of suitable calculations, explain whether the student can observe the two wavelengths as separate images. [2] angular separation = 70.624º – 70.459º = 0.165º 4 (a) Explain what is meant by a field of force. [1] (b) Describe, by means of a well-labelled diagram, the motion of electrons moving at right angles in a uniform (i) electric field, [2]

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(ii) magnetic field. [2] (c) A rectangular strip of copper of dimensions 10 mm × 5 mm × 1 mm carries a conventional current I. A magnetic field B of flux density 1.0 T is applied in a direction perpendicular to the strip as shown in Fig. 4.1.

Fig. 4.1

(i) Explain how a voltage is set up across side PQ with respect to SR. [3] (ii) Calculate the voltage, given that the velocity of the electrons is 2.52 x 10–5 m s–1. [2] 1.26 x 10–5 V

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PJC P3 1 (a) State Newton’s second law of motion. [1] (b) Fig. 1.1 shows block A of mass 4.0 kg and block B of mass 1.0 kg connected by a light cord that passes over a frictionless pulley. Block A lies on a rough plane inclined at 45° to the horizontal. The frictional force between block A and the plane is 15 N.

Fig. 1.1

(i) Determine the magnitude of the acceleration of the two blocks and the tension in the cord. [4] T = 10.4 N a = 0.587 m s−2 (ii) When block A is 1.0 m vertically above the ground, the cord breaks. The velocity of block A at that instant is 0.5 m s–1. Calculate the speed of block A just before it reaches the ground. [2] 3.04 m s−1 (c) After reaching the smooth ground, block A travels some further distance before colliding with a stationary block C of mass 6.0 kg. The velocity of block A before collision is 1.6 m s–1, as shown in Fig. 1.2.

Fig. 1.2

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(i) State the principle of conservation of momentum. [1] (ii) Upon collision, block C moves to the right with a speed of 0.70 ms–1. Calculate the speed of block A immediately after the collision. [2] 0.55 m s−1 (iii) Hence, discuss quantitatively, whether the collision between blocks A and C is elastic. [2] 2 A body M of mass 200 g moves with simple harmonic motion of amplitude 15 cm and period 2.0 s. (a) Calculate (i) its maximum speed, [1] 0.47 m s−1

(ii) its maximum acceleration, [1] 1.5 m s−2

(iii) its displacement 0.10 s after passing through its maximum displacement. 0.14 m [2] (b) (i) Show that the total energy of a body of mass m undergoing simple

harmonic motion is 12

mω2xO2. [2]

(ii) Sketch the graphs of total energy ET , kinetic energy EK and potential energy EP of the body against its displacement x from the equilibrium position. Label the graphs clearly, with appropriate quantities m, ω and xO

marked on the axes. [3]

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3 Fig. 3.1 shows a potential divider arrangement using a fixed resistor of resistance 4.0 Ω and a variable resistor of maximum resistance 20 kΩ with a slide contact connected to terminal S.

Fig. 3.1

The e.m.f. of the battery is 12 V and it has negligible internal resistance. It is possible to obtain different continuously-variable ranges by selecting, as the output, particular pairs of terminals from S, X, Y and Z. (a) (i) Calculate the voltage range obtainable between the terminals S and X. [2] 0 V – 10 V (ii) Hence, or otherwise, calculate the voltage range between the terminals S and Z. [1] 2 V – 12 V (b) The slide contact S is set at the mid-point of the 20 kΩ resistance track. A voltmeter of resistance of 10 kΩ is then connected between S and Y. Calculate the reading on the voltmeter. [3] 3.2 V

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(c) The variable resistor in Fig. 3.1 is replaced by a thermistor T, as shown in Fig. 3.2. At room temperature, the resistance of the thermistor is 12 kΩ. When it is placed in hot liquid, its resistance falls to 2.0 kΩ.

Fig. 3.2

(i) Sketch the resistance/Ω against temperature/ºC characteristic of the thermistor. [1] (ii) Using the band theory, explain the variation of the thermistor’s resistance with temperature. [3] (iii) State a practical use of the thermistor. [1]

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5 (a) The value of the gravitational potential Φ at a point in the Earth’s field is

given by the equation Φ = – GM r where M is the mass of the Earth and r is

the distance of the point from the centre of the Earth such that r is greater than the radius of the Earth RE. (i) Define gravitational potential at a point. [1] (ii) Explain why the potential has a negative value. [1] (b) Fig. 5.1 shows the equipotential lines for Earth, where point A is at a potential of − 4.0 ×107 J kg−1 and points B and C are at a potential of

− 5.0 ×107 J kg−1.

Fig. 5.1

(i) On Fig. 5.1, draw the equipotential line for the gravitational potential of − 4.5 ×107 J kg−1. [1]

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(ii) Calculate the work done by the gravitational field in bringing a body of mass 3000 kg from A to B. [2] 3.0 x 1010 J (iii) The work done by the gravitational force in bringing the mass from B to C along the equipotential line is zero. Explain why this is so. [1] (c) (i) Show that a body projected from the Earth’s surface (assumed stationary)

with a speed equal to or greater than the escape speed 2gRE will never return. State any assumption(s) made in your workings for this result to be valid. [3] (ii) Information related to the Earth and the Sun is given below.

mass of Sunmass of Earth = 3.3 x 105

radius of Sunradius of Earth = 110

Given that the escape speed from the Earth is 1.1×104 m s−1, calculate the escape speed from the Sun. [2] 6.1 x 105 m s−1 (iii) The surface temperature of the Sun is about 6000 K and hydrogen is the most abundant element in the Sun’s atmosphere. Explain why this is so by means of suitable calculations, assuming that hydrogen is an ideal gas. [2] 1.2 x 104 m s−1

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(d) Fig. 5.2 shows the way in which the gravitational potential energy of a body of mass m depends on r.

Fig. 5.2

(i) What does the gradient of the tangent to the curve at r = RE represent? [1] (ii) The body is projected vertically upwards from the Earth’s surface. At a certain distance R from the centre of the Earth, the total energy of the body may be represented by a point on the line XY. Five points, A, B, C, D, E have been marked on this line. Explain clearly which point(s) could represent the total energy of the body 1. if it were momentarily at rest at the top of its trajectory, [2] 2. if it were falling towards the Earth, [2] 3. if it were moving away form the Earth, with sufficient energy to reach an infinite distance? [2]

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6 (a) State what is meant by saying that a temperature is on an absolute scale. [1] (b) Explain what is meant by (i) an ideal gas, [1] (ii) absolute zero on the Kelvin scale, [1] (iii) the internal energy of a gas. [2] (c) A car tyre has a fixed internal volume of 0.0160 m3. On a day when the temperature is 27°C, the pressure in the tyre has to be increased from 2.76 ×105 Pa to 3.91×105 Pa. (i) Assuming that the air is an ideal gas, calculate the amount of air which has to be supplied at constant temperature. [3] 0.738 moles (ii) A portable supply of air used to inflate tyres has a volume of 0.0117 m3 and is filled with air at a pressure of 1.165 ×106 Pa. Show that, at 27°C, there is more than enough air in it to supply four tyres, as in (c)(i), without the pressure falling below 4.00 ×105 Pa. [3] 5.465 moles > 2.951 moles (iii) Show that the internal energy of a molecule of air at a temperature of 27°C is 6.21 x 10–21 J. Assume that the air behaves as a monatomic ideal gas. [2] (iv) Hence, calculate the internal energy of one mole of the air at a temperature of 27°C. [1] 3740 J

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(d) In order to study the sudden compression of a gas, some dry air is enclosed in a cylinder fitted with a piston, as shown in Fig. 6.1.

Fig. 6.1

The mass of air in the cylinder is constant. The material of the cylinder and the piston is an insulator so that no thermal energy enters or leaves the air. The volume and pressure of air are measured. The piston is then moved suddenly to compress the air and the new volume and pressure are measured. The variation with volume V of the pressure p of the air is shown in Fig. 6.2.

Fig. 6.2

It may be assumed that the dry air behaves as an ideal gas. (i) By considering the pressure and volume of the dry air at points A and B, and using the equation of state for an ideal gas, show that the temperature of the air increases when the air is compressed. [3] 480 kg m2 s–2 > 250 kg m2 s–2 (ii) The dry air then goes through two more processes. Process 1: The gas is cooled while keeping the piston at the same position. Process 2: The gas then expands, while kept at constant temperature, to return to its original state. On Fig. 6.2, draw and label the p-V graphs of the two processes described above. [3]

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RV P2 1 (a) A lecturer holds a flat S$2 note just above a student’s open fingers as shown in Fig. 1.1.

He challenges the students that whoever can catch the S$2 note when he releases it can keep it. Explain quantitatively whether any student would be able to catch the note when the lecturer releases it. [3] (b) A lecturer sees a student who owes him homework a distance d away. The lecturer immediately moves towards the student with a constant velocity vL. The student sees the lecturer moving towards him to tO seconds later and starts moving away in the same direction at a constant velocity vS. Write down an expression for the time taken t for the lecturer to catch up with the student from the instant he sees the student. Show your working clearly. [4]

t = d − vs to vL− vs

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2 Jane, whose mass is 50.0 kg, needs to swing across a river (having width D) filled with man-eating crocodiles to save Tarzan from danger. She must swing on a vine into a wind exerting a constant horizontal force F. The vine has a length L and initially makes an angle θ with the vertical (Fig. 2.1). Take D = 50.0 m, F = 110 N, L = 40.0 m, and θ = 50.0°.

(a) Show that the angle Φ is 28.9º. [1] (b) Calculate the minimum speed Jane needs to begin her swing in order for her to just reach Tarzan. [3] 6.12 m s−1

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(c) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing if Tarzan has a mass of 80.0 kg? [2] 9.90 m s−1

3 A small immersion electrical heater, operating at a constant power, was used to heat 64 g of water in a thin plastic cup. The mass of the cup was negligible. The temperature of the water was recorded at regular intervals for 30 minutes and a graph of temperature against time is drawn as shown in Fig. 3.1 below.

(a) (i) Use the graph to determine the initial rate of temperature rise of the water. 0.07 – 0.18 K s–1 [2] (ii) The specific heat capacity of water is 4200 J kg–1 K–1. Determine the rate at which energy was supplied to the water by the heater. [2] 18 – 50 W (b) After 26 minutes the rate of temperature rise became very small. Explain why. [1]

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(c) The experiment was repeated using the same mass of water in a thick ceramic mug. The initial temperature of the water was the same and the water was heated for the same length of time. (i) On Fig 3.1, sketch a possible graph of temperature against time for the water in the thick ceramic mug. [1] (ii) Explain your reasoning for your graph. [2] 4 (a) A and B are two identical conducting spheres each carrying a charge +Q. They are placed in a vacuum with their centres distance d apart as shown in Fig. 4.1.

Fig. 4.1

Explain why the force F between them is not given by the expression F = Q2

4πε0d2 .

[2] (b) Electric fields and magnetic fields may be represented by lines of force. Fig. 4.2 shows some lines of force.

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(i) State whether the field strength at the vicinity of A and at the vicinity of B is constant, increasing or decreasing when measured in the direction from A towards B. [2] (ii) Explain why the field lines can never touch or cross. [2] 5 (a) Distinguish between electromotive force and potential difference. [2] (b) An electric hotplate is designed to operate on a power supply of 240 V has two coils of wire of resistivity of 9.8 × 10–7 Ω m. Each coil of wire has a length of 16 m and a cross-sectional area 0.20 mm2. (i) For one of the coils, calculate 1. its resistance, [2] 78.4 Ω 2. the power dissipation when a 240 V supply is connected across it. [2] 735 W

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(ii) Fig. 5.1 shows how the two coils can be connected to operate at different powers.

On Fig. 5.2, fill up the table with ‘ON’ or ‘OFF’ to obtain the lowest and highest levels of operating power.

Switch A Switch B Switch C

Lowest Power Highest Power

[2] RV P3 1 (a) Fig. 1.1 shows two small dots P and P’ on a printed signboard at a distance L away from the eyes of a reader.

Fig. 1.1

The distance between the dots is s. Write down an expression for the angular separation q between the dots, in terms of L and s. [1] (b) The visual acuity α of the eye is the minimum angular separation of two equidistant points which can just be distinguished by the eye. Experiments show that most people just failed to see the division of a millimeter on a ruler when the ruler is about 2.0 m away from the observer. Estimate the visual acuity α of the average person. [2] 5 x 10−4 radian

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(c) For the average person, the least distance of distinct vision d is about 25 cm. Any nearer object will appear blur to the person. Estimate the thickness of the finest line of a printed drawing that can be distinguished clearly by an average reader. [2] 0.125 mm (d) Some laser printers have a print resolution of 500 dots per inch. If a line consisting of alternate white and black dots is printed by such a printer, can the individual dots be distinguished? [2] [1 inch = 25.4 mm] 4.08 x 10−4 radian 2 (a) Define moment of a force. [1] (b) A person supports a load of 20 N in his hand as shown in Fig 2.1. The system of the hand and load is represented by Fig 2.2. The rod represents the forearm and T represents the tension exerted in the biceps. The forearm weighs 65 N.

(i) Show that the tension T in the biceps is 410 N. [2] (ii) Determine the magnitude and direction of the force acting at the elbow. [4] Rx = 140.4 N Ry = 300.7 N (c) A karate expert can split a stack of bricks by bringing her arm and hand swiftly against the bricks with considerable speed. Using Newton’s laws of motion, explain why she has to execute the karate strike very quickly. [4]

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3 A mass P, 80 g, is attached to the free end of a horizontal spring on a smooth surface. The spring-mass system is set into simple harmonic motion by pulling P to the right of the equilibrium position and is released from rest as shown in Fig. 3.1.

If the air resistance on P is negligible, the variation of the velocity v of P with displacement x is shown in Fig. 3.2. Vectors to the right are taken to be positive.

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(a) For the motion of P, determine (i) the amplitude, [1] 40 mm (ii) the frequency. [2] 2.4 Hz (b) If the air resistance on P is not negligible, sketch on Fig. 3.2 the variation of the velocity of P with displacement x. [2] (c) A periodic force is now exerted on the spring-mass system. When the periodic force is at a certain frequency, P is in resonance. (i) Explain what is meant by the term resonance. [1] (ii) 1. Using energy consideration, explain why the total energy of the system increases to another value at steady state. [2] 2. Given that the total energy of the spring-mass system at steady state is doubled. Determine the new maximum speed of P. [2] 0.85 m s−1

(iii) On Fig. 3.2, sketch the variation of the velocity of P, at resonance, with displacement x. [2] Maximum speed at 0.85 m s-1 and amplitude at 57 mm. 4 Fig. 4 shows a pair of identical loudspeakers A and B placed 2.00 m apart and emitting coherent sound waves of frequency 470 Hz. An observer walks from X to Y. The perpendicular distance between the sources and XY is 12.0 m. As he walks, he hears sound of maximum intensity at P, followed by minimum intensity at Q and the next maximum intensity at R. R is 4.50 m away from P.

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Fig. 4

(a) Explain why the observer hears sound of maximum and minimum intensity as he moves from X to Y. [2] (b) (i) AR is 12.5 m, show that BR is 13.2 m to 3 significant figures. [1] (ii) Determine the wavelength of the sound. [2] 0.700 m (iii) Determine the speed of the sound. [2] 329 m s−1

(c) The power of the loudspeakers A and B are identical. Suggest why the intensity at Q is not zero. [3]

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5 (a) Define gravitational potential at a point in a gravitational field and state its unit. [2] (b) Fig. 5.1 shows the variation of gravitational potential between the surface of Moon and the surface of Earth along the line joining the centres.

Fig. 5.1

The following data is required in answering the question. mass of Earth 5.98 x 1024 kg mass of Moon 7.35 x 1022 kg distance from the centre of Moon to the centre of Earth 3.84 x 108 m (i) State how the resultant gravitational field strength can be deduced from Fig. 5.1. [2] (ii) State the gravitational field at point P. [1]

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(iii) Hence, or otherwise, determine distance X. [3] X = 3.46×10 m (iv) A rocket of mass 2.7 × 106 kg on a mission to the Moon is to be launched from Earth. In order to reach the surface of the Moon, the rocket must be launched with a minimum speed. 1. Using Fig. 5.1, determine this minimum speed. Explain your working clearly. [4] 1.10×10 m s–1 2. With this minimum speed, calculate the speed at which the rocket will land on the Moon’s surface. [2] 2.28×10 m s–1 (c) The Moon is a natural satellite of the Earth. It can be assumed that the Moon travels at a constant speed around the Earth in a circular path, with the Earth at the centre of the circle. (i) Using Newton’s laws of motion, explain why an object travelling in a circle with constant speed has acceleration. State the direction of this acceleration. [3] (ii) Show that orbital period T of a satellite and its distance r from the Earth is

given by T2 = 4π2

GME r3. [3]