AHL 9.1 Energy transformation

Transcription

1 AHL 9.1 Energy transformation [1 mark] A pendulum oscillating near the surface of the Earth swings with a time period T. What is the time period of the same pendulum near the surface of the planet Mercury where the gravitational field strength is 0.4g? A. 0.4T B. 0.6T C. 1.6T D. 2.5T C 2. [1 mark] A mass oscillates with simple harmonic motion (SHM) of amplitude x o. Its total energy is 16 J. What is the kinetic energy of the mass when its displacement is? A. 4 J B. 8 J C. 12 J D. 16 J C 1

2 3. [1 mark] A particle is oscillating with simple harmonic motion (shm) of amplitude x 0 and maximum kinetic energy E k. What is the potential energy of the system when the particle is a distance 0.20x 0 from its maximum displacement? A. 0.20E k B. 0.36E k C. 0.64E k D. 0.80E k C 4. [1 mark] A mass is connected to a spring on a frictionless horizontal surface as shown. The spring is extended beyond its equilibrium length and the mass executes simple harmonic motion (SHM). Which of the following is independent of the initial displacement of the spring? A. The angular frequency of the oscillation B. The total energy of the mass C. The average speed of the mass D. The maximum kinetic energy of the mass 2

3 A 5. [1 mark] The period of a particle undergoing simple harmonic motion (SHM) is. The ratio is proportional to A.. B.. C.. D. A 6. [1 mark] A particle of mass oscillates with simple harmonic motion (SHM) of angular frequency. The amplitude of the SHM is. What is the kinetic energy of the particle when it is half way between the equilibrium position and one extreme of the motion? A. B. C. D. 3

4 B 7. [1 mark] The bob of a pendulum has an initial displacement 0 to the right. The bob is released and allowed to oscillate. The graph shows how the displacement varies with time. At which point is the velocity of the bob at maximum towards the right? A 8. [1 mark] A particle undergoes simple harmonic motion (SHM) of maximum kinetic energy E max and amplitude x 0. The particle is released from rest at its maximum displacement amplitude. What is the change in the kinetic energy when the particle has travelled a distance of? A. 4

5 B. C. D. C The candidates found this question to be the most difficult of the paper, with the correct answer being the least often selected! The key to spotting the correct solution is a simple diagram showing that after the particle has travelled a distance of x0/3 then its distance to the equilibrium position is 2x0/3. Substituting this value into the relevant equation in the Data Booklet gives response C directly. 9. [1 mark] A body moves with simple harmonic motion (SHM) with period T and total energy E T. What is the total energy when the period of the motion is changed to 5T and the amplitude of the motion remains constant? A E T B. 0.2 E T C. 5 E T D. 25 E T A 10. [1 mark] A small point mass m is placed at the same distance from two identical fixed spherical masses far from any other masses. 5

6 The point mass is released from rest. The point mass will A. move upwards. B. stay where it is. C. move towards P and stop there. D. oscillate about point P. D 11. [1 mark] A particle P executes simple harmonic motion (SHM) about its equilibrium position Y. The amplitude of the motion is XY. 6

7 At which of the positions shown on the diagram is the acceleration of P equal to zero and the kinetic energy of P equal to zero? A 12. [1 mark] Which graph shows how velocity v varies with displacement x of a system moving with simple harmonic motion? A 7

8 This question was very poorly answered. There are many graphs associated with simple harmonic motion (SHM) which are sinusoidal, but these are the graphs with time on the horizontal axis. Having displacement on the axis, though, will produce different graphs and candidates should be equally familiar with these. In this case it should have been clear that at the extremities of SHM velocity will be zero, while at the equilibrium point it will be maximum. So the only possible answer is A, showing half a cycle of SHM. 13. [1 mark] An object undergoes simple harmonic motion. Which graph shows the relationship between the acceleration a and the displacement x from the equilibrium position? A 14. [1 mark] An object undergoes simple harmonic motion (SHM). The total energy of the object is proportional to A. the amplitude of the oscillations. B. the time period of the oscillations. C. the frequency of the oscillations. 8

9 D. the mass of the object. D 15. [1 mark] A particle is undergoing simple harmonic motion (SHM) in a horizontal plane. The total mechanical energy of the system is E. Which of the following correctly gives the kinetic energy of the particle at the positions of maximum displacement and equilibrium? B 16. [1 mark] The equation for the velocity of an object performing simple harmonic motion is of the following is a correct alternative form of the equation?. Which A. B. 9

10 C. D. A 17a. [2 marks] This question is about simple harmonic motion (SHM). The graph shows the variation with time of the acceleration of an object X undergoing simple harmonic motion (SHM). Define simple harmonic motion (SHM). 10

11 force/acceleration proportional to the displacement/distance from a (fixed/equilibrium) point/mean position; directed towards this (equilibrium) point / in opposite direction to displacement/ distance; Allow algebra only if symbols are fully explained. A good definition of simple harmonic motion must focus on the proportionality between acceleration and displacement from some fixed point and on the directional relationship between acceleration and displacement. Many failed to emphasise the fixed-point aspect of the definition. Attempts made to define simple harmonic motion in algebraic terms normally omitted a clear statement of the symbols and the meaning of the negative sign. 17b. [1 mark] X has a mass of 0.28 kg. Calculate the maximum force acting on X. 11

12 0.73 (N); (allow answer in range of 0.71 to 0.75 (N)) This was almost universally well done. 17c. [4 marks] Determine the maximum displacement of X. Give your answer to an appropriate number of significant figures. use of ; to (rad s 1 )) or or ; } (allow answers in the range of T = 7.8 to 8.0 (s) or = 12

13 ; (allow answers in the range of 4.0 to 4.25 (m)) two significant figures in final answer whatever the value; Award [4] for a bald correct answer. Similarly, this was well done. Most appreciated the need to truncate the final answer to a sensible number of significant digits (two in this case). 17d. [2 marks] A second object Y oscillates with the same frequency as X but with a phase difference of using the graph opposite, how the acceleration of object Y varies with.. Sketch, shape correct, constant amplitude for new curve, minimum of 10 s shown; } (there must be some consistent lead or lag and no change in T) lead/lag of 1 s (to within half a square by eye); 13

14 A large number of candidates could not translate a phase change into the correct time lead or lag. Most used the answer for were acceptable. and lost a mark in consequence. Most free-hand sketches of sine curves 18a. [2 marks] This question is about simple harmonic motion (SHM An object is placed on a frictionless surface. The object is attached by a spring fixed at one end and oscillates at the end of the spring with simple harmonic motion (SHM). The tension F in the spring is given by F = k x where x is the extension of the spring and k is a constant. Show that. 14

15 ma kx; ; (condone lack of negative sign) or implied use of defining equation for simple harmonic motion ; so ; 18b. [3 marks] One cycle of the variation of displacement with time is shown for two separate mass spring systems, A and B. (i) Calculate the frequency of the oscillation of A. (ii) The springs used in A and B are identical. Show that the mass in A is equal to the mass in B. 15

16 (i) (Hz); (ii) frequency/period is the same so ω is the same; k is the same (as springs are identical); (so m is the same) 18c. [5 marks] The graph shows the variation of the potential energy of A with displacement. 16

17 On the axes, (i) draw a graph to show the variation of kinetic energy with displacement for the mass in A. Label this A. (ii) sketch a graph to show the variation of kinetic energy with displacement for the mass in B. Label this B. 17

18 (ii) end displacements correct 0.01m; maximum lower than 0.16J; maximum equal to 0.04J half square; 19a. [3 marks] This question is about the oscillation of a mass. A mass of 0.80 kg rests on a frictionless surface and is connected to two identical springs both of which are fixed at their other ends. A force of N is required to extend or compress each spring by 1.0 mm. When the mass is at rest in the centre of the arrangement, the springs are not extended. The mass is displaced to the right by 60 mm and released. 18

19 Determine the acceleration of the mass at the moment of release. force of 1.8 N for each spring so total force is 3.6 N; acceleration ; (allow ECF from first marking point) to left/towards equilibrium position / negative sign seen in answer; This is a slightly different situation. Most candidates at SL did not use F and m to find acceleration. Very few added the force due to each spring and ECF was frequently applied. 19b. [2 marks] Outline why the mass subsequently performs simple harmonic motion (SHM). 19

20 force/acceleration is in opposite direction to displacement/towards equilibrium position; and is proportional to displacement; 19c. [2 marks] Calculate the period of oscillation of the mass. 20

21 ; ; Watch out for ECF from (a)(i) eg award [2] for for. 19d. [2 marks] The motion of an ion in a crystal lattice can be modelled using the mass spring arrangement. The interatomic forces may be modelled as forces due to springs as in the arrangement shown. The frequency of vibration of a particular ion is and the mass of the ion is. The amplitude of vibration of the ion is. Estimate the maximum kinetic energy of the ion. 21

22 ; ; Allow answers in the range of 4.8 to if 2 sig figs or more are used. 19e. [3 marks] On the axes, draw a graph to show the variation with time of the kinetic energy of mass and the elastic potential energy stored in the springs. You should add appropriate values to the axes, showing the variation over one period. 22

23 KE and PE curves labelled very roughly and shapes; } (allow reversal of curve labels) KE and PE curves in anti-phase and of equal amplitude; at least one period shown; either marked correctly on energy axis, or marked correctly on time axis; Care was needed in showing the constant and equal amplitudes. Many poor answers were seen. 19f. [1 mark] Calculate the wavelength of an infrared wave with a frequency equal to that of the model in (b). is equivalent to wavelength of ; 23

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