1. A toy locomotive of mass 0.50kg is initially at rest on a horizontal track. The locomotive is powered by a twisted rubber band which, as it unwinds, exerts a force which varies with time as shown in the table. time/s 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 force/n 0.20 0.18 0.15 0.12 0.10 0.08 0.05 0.02 0.00 Polesworth High School 1
(a) On the grid below plot a graph of force against time for the rubber band power source. force/n time/s Polesworth High School 2
State what is given by the area between the graph and the time axis. (4) (b) The rubber band is wound up and released to power the locomotive. Use your graph to show that the speed of the locomotive 8.0s after the twisted rubber band is released is l.6 m s 1. Ignore the effects of air resistance and energy losses due to friction............. (2) (c) 8.0s after release the locomotive collides with and couples to a toy truck, initially at rest, which has a mass of 1.50kg. Calculate the speed of the coupled locomotive and truck after collision. Calculate the combined kinetic energy of the locomotive and truck immediately after collision. (iii) Show, with the aid of a calculation, whether or not the collision is elastic. (5) (Total 11 marks) Polesworth High School 3
2. A 10 µf capacitor is connected across the terminals of a 100 V d.c. power supply and allowed to charge fully. (a) Calculate the charge on the capacitor, the energy stored by the capacitor. (2) (b) The fully-charged capacitor is disconnected from the power supply and connected via two wires across the terminals of an uncharged 10 µf capacitor as shown. 10µ f 10µ f The charge on the original 10 µf capacitor is shared equally between the capacitors in the parallel combination. Calculate the potential difference across the terminals of each capacitor. Calculate the total energy stored by the two capacitors. Polesworth High School 4
(iii) Account for the difference between the energy stored by the two capacitors in parallel and that stored by the original single 10 µf capacitor. (4) (Total 6 marks) 3. The diagram shows an arrangement in a vacuum to deflect protons into a detector using a magnetic field, which can be assumed to be uniform within the square shown and zero outside it. The motion of the protons is in the plane of the paper. detector path of proton proton magnetic deflector (a) Sketch the path of a proton through the magnetic deflector. At any point on this path draw an arrow to represent the magnetic force on the proton. Label this arrow F. (2) (b) State the direction of the uniform magnetic field causing this motion.... (1) Polesworth High School 5
(c) The speed of a proton as it enters the deflector is 5.0 10 6 ms 1. If the flux density of the magnetic field is 0.50T, calculate the magnitude of the magnetic force on the proton.......... (2) (d) If the path were that of an electron with the same velocity, what two changes would need to be made to the magnetic field for the electron to enter the detector along the same path?...... (2) (Total 7 marks) 4. (a) Define electric field strength, and state whether it is a scalar quantity or a vector quantity. Complete the diagram below to show the electric field lines in the region around two equal positive point charges. Mark with a letter N the position of any point where the field strength is zero. + + (6) (b) Point charges A, of +2.0 nc, and B, of 3.0 nc, are 200 mm apart in a vacuum, as shown by the figure. The point P is 120 mm from A and 160 mm from B. Polesworth High School 6
P 160 mm B 3.0nC 120 mm 200 mm A +2.0nC Calculate the component of the electric field at P in the direction AP. Calculate the component of the electric field at P in the direction PB. (iii) Hence calculate the magnitude and direction of the resultant field at P. (6) Polesworth High School 7
(c) Explain why there is a point X on the line AB in part (b) at which the electric potential is zero. Calculate the distance of the point X from A. (4) (Total 16 marks) 5. (a) State, in words, Newton s law of gravitation................... (2) Polesworth High School 8
(b) Some of the earliest attempts to determine the gravitational constant, G, were regarded as experiments to weigh the Earth. By considering the gravitational force acting on a mass at the surface of the Earth, regarded as a sphere of radius R, show that the mass of the Earth is given by M = gr 2, G where g is the value of the gravitational field strength at the Earth s surface............. (2) (c) In the following calculation use these data. radius of the Moon = 1.74 10 6 m gravitational field strength at Moon s surface = 1.62 N kg 1 mass of the Earth M = 6.00 10 24 kg gravitational constant G = 6.67 10 11 Nm 2 kg 2 Calculate the mass of the Moon and express its mass as a percentage of the mass of the Earth................ (3) (Total 7 marks) Polesworth High School 9
6. A parallel plate capacitor consists of two smooth square metal plates with sides of length 0.35 m. The gap between the plates is filled completely with a sheet of mica of thickness 0.15mm. The plates of the capacitor are connected across the terminals of a 1.2 kv supply. Refer to the Data Sheet for any additional information. (a) Calculate the magnitude of the charge on the capacitor, the resistance of the mica sheet, i.e. the resistance between the plates. (6) (b) When the capacitor is disconnected from the supply there is a leakage current through the mica sheet and the charge reduces to zero in about 3 hours. Calculate the mean current flowing during the loss of charge from the capacitor.......... (2) Polesworth High School 10
(c) Explain why there is a limit to the magnitude of the potential difference that may be applied between the plates of the capacitor............. (3) (Total 11 marks) 7. A camera flashgun uses the discharge of a capacitor to provide the energy to produce a single flash. In a particular flashgun a 4700 µf capacitor is initially charged from a 90 V supply. (a) Calculate the charge stored by the capacitor when it is fully charged, the energy stored by the fully-charged capacitor, (iii) the average current which flows if total discharge of the capacitor takes place effectively in 30 ms. (3) Polesworth High School 11
(b) During a partial discharge of the capacitor the potential difference between its terminals falls from 90 V to 80 V. Calculate the energy discharged to the flashgun.......... (2) (Total 5 marks) 8. (a) Give an equation showing how the principle of conservation of momentum applies to the colliding snooker balls shown in the diagram. u 1 2 u m 1 2 m v 1 2 v m 1 2 m State the condition under which the principle of conservation of momentum applies. (3) Polesworth High School 12
(b) A trolley, A, of mass 0.25 kg and a second trolley, B, of mass 0.50 kg are held in contact on a smooth horizontal surface. A compressed spring inside one of the trolleys is released and they then move apart. The speed of A is 2.2 m s 1. Calculate the speed of B. Calculate a minimum value for the energy stored in the spring when compressed. (4) (c) The rotor blades of a helicopter sweep out a cross-sectional area, A. The motion of the blades helps the helicopter to hover by giving a downward velocity, υ, to a cylinder of air, density ρ. The cylinder of air has the same cross-sectional area as that swept out by the rotor blades. Explaining your reasoning, derive an expression for the mass of air flowing downwards per second, and derive an expression for the momentum given per second to this air. Polesworth High School 13
(iii) Hence show that the motion of the air results in an upward force, F, on the helicopter given by F = ρaυ 2. (5) (d) A loaded helicopter has a mass of 2500 kg. The area swept out by its rotor blades is 180m 2. If the downward flow of air supports 50% of the weight of the helicopter, what speed must be given to the air by the motion of the rotor blades when the helicopter is hovering? Take the density of air to be 1.3 kg m 3.......... (3) (Total 15 marks) 9. P A 250 mm B A 150 g mass is attached to one end of a light inextensible string and the other end of the string is fixed at a point P as shown in the diagram above. The mass is held at point A so that the string is taut and horizontal. The mass is released so that it moves freely along a circular arc of 250 mm radius. Polesworth High School 14
When the string moves through the vertical position, the mass is at point B. Neglecting the effect of air resistance, calculate the kinetic energy of the mass,............ the velocity of the mass,...... (iii) the centripetal force acting on the mass,...... (iv) the tension in the string....... (Total 6 marks) 10. (a) A capacitor is made from two parallel metal plates of the same area, separated by an air gap. It is connected across a battery of constant e.m.f. The plates are moved further apart, maintaining the same area of overlap, whilst the battery remains connected. State and explain what change, if any, occurs to the potential difference across the plates, Polesworth High School 15
the capacitance of the capacitor, (iii) the charge on each plate of the capacitor, (iv) the energy stored by the capacitor. (4) (b) A thunder cloud and the earth beneath it can be considered to form a parallel plate capacitor. The area of the cloud is 8.0 km 2 and it is 0.75 km above the earth. Calculate the energy stored if the potential difference between the cloud and the earth is 200 kv. The air suddenly conducts, allowing all the charge to flow to earth in 120 µs. Calculate the mean current flowing between the cloud and the earth when this happens. (6) (Total 10 marks) Polesworth High School 16
11. The gravitational field strength at the surface of a planet, X, is 19 N kg 1. (a) Calculate the gravitational potential difference between the surface of X and a point 10 m above the surface, if the gravitational field can be considered to be uniform over such a small distance. Calculate the minimum amount of energy required to lift a 9.0 kg rock a vertical distance of 10m from the surface of X. (iii) State whether the minimum amount of energy you have found in part (a) would be different if the 9.0 kg mass were lifted a vertical distance of 10 m from a point near the top of the highest mountain of planet X. Explain your answer. (3) (b) Calculate the gravitational field strength at the surface of another planet, Y, that has the same mass as planet X, but twice the diameter of X............. (2) (Total 5 marks) Polesworth High School 17
12. An experiment is performed to investigate the magnetic field inside a 100 mm long solenoid of 500 turns. A small single loop of wire attached to a voltage recorder (data logger) is lowered coaxially inside the solenoid, as shown in the diagram, until it is at the centre of the solenoid. voltage recorder lowered inside single loop power supply solenoid The solenoid is supplied with a steady current of 0.50 A. The magnetic flux density at a point on the axis well inside a long solenoid is given by µ B = 0 NI l, where N is the number of turns on the solenoid and l is its length. µ 0 and I have their usual meanings. Calculate the approximate value of the flux density at the centre of the solenoid on its axis....... The single loop of wire is positioned at the centre of the solenoid so that it is at right angles to the magnetic field. If the loop has an area of 160 mm 2, calculate the magnetic flux through the loop. (Total 3 marks) Polesworth High School 18
13. Take the acceleration due to gravity, g E, as 10 m s 2 on the surface of the Earth. The acceleration due to gravity on the surface of the Moon is g E. An object whose weight on 6 Earth is 5.0 N is dropped from rest above the Moon s surface. What is its momentum after falling for 3.0s? A 2.5 kg m s 1 B 6.2 kg m s 1 C 15 kg m s 1 D 25 kg m s 1 (Total 1 mark) Polesworth High School 19
14. X l A simple pendulum consists of a bob of mass m on the end of a light string of length l. The bob is released from rest at X when the string is horizontal. When the bob passes through Y its velocity is υ and the tension in the string is T. Which one of the following equations gives the correct value of T? Y A T = mg B T = mυ 2 l C T + mg = mυ 2 l D T mg = mυ 2 l (Total 1 mark) Polesworth High School 20
15. In experiments to pass a very high current through a gas, a bank of capacitors of total capacitance 50 µf is charged to 30 kv. If the bank of capacitors could be discharged completely in 5.0 ms what would be the mean power delivered? A B C D 9.0 MW 4.5 MW 110 kw 22 kw (Total 1 mark) 16. (a) Give an equation showing how the principle of conservation of momentum applies to the colliding snooker balls shown in the diagram. u 1 2 u m 1 2 m v before collision 1 2 v m 1 2 m after collision State the condition under which the principle of conservation of momentum applies. (3) Polesworth High School 21
(b) A trolley, A, of mass 0.25 kg and a second trolley, B, of mass 0.50 kg are held in contact on a smooth horizontal surface. A compressed spring inside one of the trolleys is released and they then move apart. The speed of A is 2.2 m s 1. Calculate the speed of B. Calculate a minimum value for the energy stored in the spring when compressed. (4) (c) The rotor blades of a helicopter sweep out a cross-sectional area, A. The motion of the blades helps the helicopter to hover by giving a downward velocity, υ, to a cylinder of air, density ρ. The cylinder of air has the same cross-sectional area as that swept out by the rotor blades. Explaining your reasoning, derive an expression for the mass of air flowing downwards per second, and Polesworth High School 22
derive an expression for the momentum given per second to this air. (iii) Hence show that the motion of the air results in an upward force, F, on the helicopter given by F = ρaυ 2. (5) (d) A loaded helicopter has a mass of 2500 kg. The area swept out by its rotor blades is 180 m 2. If the downward flow of air supports 50% of the weight of the helicopter, what speed must be given to the air by the motion of the rotor blades when the helicopter is hovering? Take the density of air to be 1.3 kg m 3.......... (3) (Total 15 marks) 17. (a) State what is meant by a free vibration, Polesworth High School 23
a forced vibration. (2) (b) A car and its suspension can be treated as a simple mass-spring system. When four people of total weight 3000 N get into a car of weight 6000 N, the springs of the car are compressed by an extra 50 mm. Calculate the spring constant, k, of the system. Show that, when the system is displaced vertically and released, the time period of the oscillations is 0.78 s. (3) (c) The loaded car in part (b) travels at 20 ms 1 along a road with humps spaced 16 m apart. Calculate the time of travel between the humps. Hence, state and explain the effect the road will have on the oscillation of the car. (3) (Total 8 marks) Polesworth High School 24
18. A metal aircraft with a wing span of 42m flies horizontally with a speed of 1000 km h 1 in a direction due east in a region where the vertical component of the flux density of the Earth s magnetic field is 4.5 10 5 T. Calculate the flux cut per second by the wings of the aircraft............. Determine the magnitude of the potential difference between the wing tips, stating the law which you are applying in this calculation.......... (iii) What would be the change in the potential difference, if any, if the aircraft flew due west?...... (Total 6 marks) Polesworth High School 25
19. (a) For a capacitor of capacitance C, sketch graphs of charge, Q, and energy stored, E, against potential difference, V. Q E V graph A graph B V What is represented by the slope of graph A?... (3) (b) A capacitor of capacitance 0.68 F is charged to 6.0 V. Calculate the charge stored by the capacitor,...... the energy stored by the capacitor....... (2) (Total 5 marks) 20. A chemical centrifuge consists of two test-tube holders which can be spun round in a horizontal circular path at very high speed as shown. The centrifuge runs at a steady speed of 3000 revolutions per minute and the test-tube holders are horizontal. 95 mm axis of rotation test-tube holders Polesworth High School 26
Calculate the angular speed of the centrifuge in rad s 1...... Calculate the magnitude of the acceleration at a point on the centrifuge 95 mm from the axis of rotation....... (iii) State the direction of the acceleration in part.... (Total 5 marks) 21. (a) A vibrating system which is experiencing forced vibrations may show resonance. Explain what is meant by forced vibrations...... resonance............ (3) (b) Explain what is meant by damping. Polesworth High School 27
What effect does damping have on resonance? (2) (Total 5 marks) 22. Both gravitational and electric field strengths can be described by similar equations written in the form bc a = d 2. (a) Complete the following table by writing down the names of the corresponding quantities, together with their SI units, for the two types of field. symbol gravitational field quantity SI unit quantity electrical field SI unit a gravitational field strength b c d 1 4πε 0 m F 1 (4) (b) Two isolated charged objects, A and B, are arranged so that the gravitational force between them is equal and opposite to the electric force between them. The separation of A and B is doubled without changing their charges or masses. State and explain the effect, if any, that this will have on the resultant force between them. Polesworth High School 28
At the original separation, the mass of A is doubled, whilst the charge on A and the mass of B remain as they were initially. What would have to happen to the charge on B to keep the resultant force zero? (3) (Total 7 marks) 23. The diagram shows a current-carrying conductor which is free to move between the poles of a horseshoe magnet. The conductor is a brass rod resting on two conducting parallel rails. N S brass rod rails (a) Draw an arrow on the diagram to show the direction of the force on the rod due to the magnetic field when the switch is closed. Label it F. (1) Polesworth High School 29
(b) Using the information below, calculate the magnitude of the force F. The e.m.f. of the cell is 2.0 V. The flux density of the magnetic field in the region of the rod is 0.080 T. The total resistance of the circuit is 0.40 Ω. The rails are 60 mm apart.......... (2) (Total 3 marks) 24. A student is investigating the effect of capacitors in a.c. circuits. He connects up the circuit shown in the diagram. The variable frequency signal generator supplies a sinusoidal a.c. output at constant r.m.s. voltage. C is a capacitor whose capacitance, C, may be altered. A signal generator a.c. ammeter C (a) The student measures the r.m.s. current for several different frequencies when C is constant. He then plots graphs of current against frequency. Sketch, using the axes below, the graph you would expect him to obtain. Label this graph F. Polesworth High School 30
Also sketch, using the same axes, the graph expected when the capacitance is half the value in part. Label it S. current 0 0 frequency (iii) Show how the value of the capacitance of C can be calculated from the gradient of the graph and the r.m.s. voltage of the signal generator output..... (5) (b) The output from the signal generator was 7.8 V r.m.s. At a particular frequency the r.m.s. current was 90 ma. Calculate the reactance of the capacitor at this frequency.......... (2) (Total 7 marks) Polesworth High School 31
25. A spring, which obeys Hooke s law, hangs vertically from a fixed support and requires a force of 2.0 N to produce an extension of 50mm. A mass of 0.50kg is attached to the lower end of the spring. The mass is pulled down a distance of 20mm from the equilibrium position and then released. (a) Show that the time period of the simple harmonic vibrations is 0.70 s. Sketch the displacement of the mass against time, starting from the moment of release and continuing for two oscillations. Show appropriate time and distance scales on the axes. (5) (b) The mass-spring system described in part (a) is attached to a support which can be made to vibrate vertically with a small amplitude. Describe the motion of the mass-spring system with reference to frequency and amplitude when the support is driven at a frequency of 0.5 Hz, Polesworth High School 32
1.4 Hz. (3) (Total 8 marks) 26. (a) Figure 1 shows a pair of parallel metal plates, A and B, fixed vertically 20 mm apart with a potential difference of 1500 V between them. 20 mm P figure 1 0 V +1500 V A B Draw the electric field lines in the space between the plates and calculate the electric field strength at P. Polesworth High School 33
Sketch a graph showing the potential at different points in the space between the plates. potential 0 V A B distance (5) (b) Figure 2 shows a polystyrene ball of mass 5.0 10 4 kg suspended midway between the plates by a long insulating thread. The ball has a conducting surface and carries an initial charge of 3.0 nc. 3.0nC figure 2 20 mm 0 V +1500 V Polesworth High School 34
Calculate the force on the ball due to the electric field and hence show that the ball, when released from rest, will take a time of approximately 0.2 s to reach one conducting plate. State the direction of motion of the ball. Explain why the ball will subsequently shuttle backwards and forwards between the plates. (6) (c) The same charged ball described in part (b) is suspended at rest between the poles of a magnet. State the magnitude and direction of the force (if any) on the ball and explain your answer.......... (2) (Total 13 marks) Polesworth High School 35
27. (a) The graph shows how the gravitational potential varies with distance in the region above the surface of the Earth. R is the radius of the Earth, which is 6400 km. At the surface of the Earth, the gravitational potential is 62.5 MJ kg 1 0 distance from centre of Earth 0 R 2R 3R 4R gravitational potential/mj kg 1 20 40 60 80 Use the graph to calculate the gravitational potential at a distance 2R from the centre of the Earth, the increase in the potential energy of a 1200kg satellite when it is raised from the surface of the Earth into a circular orbit of radius 3R. (4) Polesworth High School 36
(b) Write down an equation which relates gravitational field strength and gravitational potential. By use of the graph in part (a), calculate the gravitational field strength at a distance 2R from the centre of the Earth. (iii) Show that your result for part (b) is consistent with the fact that the surface gravitational field strength is about 10 N kg 1. (5) (Total 9 marks) Polesworth High School 37
28. The curve on the graph is the measured frequency response of the audio-frequency filter shown in the circuit diagram. A fixed voltage of 5.0V r.m.s. is applied across the input of the filter using a sinusoidal signal generator of neglible internal resistance, and the r.m.s. output voltage V of the filter is measured using an oscilloscope. C 5.0 V r.m.s. R V out 5 4 r.m.s. output voltage /V V out 3 2 1 0 100 200 300 500 1k 2k 3k 5k 10k 20k frequency f/hz (a) Explain, without writing down any formulae, why the response curve takes this form................ (3) Polesworth High School 38
(b) A current of 0.5 ma r.m.s. flows in the circuit when the frequency of the supply is 20 khz. Estimate the resistance of the resistor R....... (1) (c) By considering the value of V at a frequency of 900 Hz, show that the capacitance of the capacitor C is approximately 10 nf................ (3) (d) Using the frequency response curve of the filter, estimate the frequency at which the power dissipated by the resistor R is half its maximum value............. (2) (Total 9 marks) 29. The following data refer to two planets. radius/km density/kg m 3 planet P 8000 6000 planet Q 16000 3000 Polesworth High School 39
The gravitational field strength at the surface of P is 13.4 N kg 1. What is the gravitational field strength at the surface of Q? A 3.4 N kg 1 B 13.4 N kg 1 C 53.6 N kg 1 D 80.4 N kg 1 (Total 1 mark) 30. A body is in simple harmonic motion of amplitude 0.50 m and period 4π seconds. What is the speed of the body when the displacement of the body is 0.30 m? A 0.10 m s 1 B 0.15 m s 1 C 0.20 m s 1 D 0.40 m s 1 (Total 1 mark) 31. Which one of the following statements always applies to a damping force acting on a vibrating system? A B C D It is in the same direction as the acceleration. It is in the same direction as the displacement. It is in the opposite direction to the velocity. It is proportional to the displacement. (Total 1 mark) Polesworth High School 40