shm 2012

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SIMPLE HARMONIC MOTION - 2012 Simple harmonic motion is defined as an oscillatory motion of a particle whose acceleration is directly proportional to its displacement from an equilibrium position and this acceleration is always directed towards the equilibrium position. Quantity Equations with respect to t Equation with respect to x Displacement x = x0 sin ωt (starts from

equilibrium position)

x = x0 cos ωt (starts from amplitudinal position)

x

Velocity v = ωx0 cos ωt (starts from equilibrium position)

v = – ωx0 sin ωt (starts from

amplitudinal position)

v = ±ω (x02 – x 2)

Acceleration a = – ω2 x0 sin ωt (starts from equilibrium position)

a =–ω2 x0 cos ωt (starts from

amplitudinal position)

a = – ω2 x

Kinetic Energy ½ mω2x0 2 cos2 ωt (starts from


½ mω2x0 2 sin2 ωt (starts from


½ mω2 (x0 2 – x2)

Potential Energy ½ mω2x0 2 sin2 ωt (starts from


½ mω2x0 2 cos2 ωt (starts from


½ mω2x 2

Total Energy ½ mω2x0 2 ½ mω2x0

2

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Damped Oscillation

A damped oscillation is an oscillation which occurs in the presence of dissipative forces. The total energy of the oscillating system and the amplitude of the oscillation decreases with time. Light damping - Particle undergo several oscillations before the amplitude is reduced to zero. Heavy damping - Particle does not oscillate and returns to equilibrium position after a long time. Critical damping - Particle does not oscillate and returns to equilibrium position in the shortest time. Forced Oscillation is an oscillation which occurs under the influence of an external periodic driving force and the oscillating frequency adopts the driving frequency of the external driving force. Free Oscillation is an oscillation whereby the oscillating system oscillates at its natural frequency without the influence of any external force. Hence, there is no loss or gain of energy from the surroundings. Equilibrium position is the position where the net force acting on the system is zero.

Resonance occurs when the driving frequency of the external oscillatory system is equal to the natural frequency of the oscillating system. Maximum energy is transferred to the oscillating system and it oscillates with maximum amplitude.

Amplitude, x0 , of a body in SHM is the magnitude of the maximum displacement of the body from its equilibrium position in either direction. [metre; scalar] Displacement, x, of a body in SHM is the distance from its equilibrium position at a particular time. [metre; vector] Period, T, of a body in SHM is the time taken for 1 complete oscillation of the body. [seconds; scalar] Angular frequency, ω, of a body in SHM is the number of completed cycles (in terms of angular displacement) per unit time of the body, whereby 1 completed cycle represents 2π radians. It is also the magnitude of angular velocity. [rad s-1; scalar] Phase is defined as the stage of oscillation that has been completed at a specific reference time in terms of an angle. Phase difference, θ, between two bodies in SHM is the delay or advance in the start of one body’s cycle with respect to another, expressed in terms of an angle.

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SHM MCQS 1. A particle oscillates with undamped simple harmonic motion. Which one of the following statements about the acceleration of the oscillating particle is true? A) It is least when the speed is greatest. B) It is always in the opposite direction to its velocity. C) It is proportional to the frequency. D) It decreases as the potential energy increases. 2. Which one of the following gives the phase difference between the particle velocity and the particle displacement in simple harmonic motion? A) π/4 rad B) π/2 rad C) 3π/4 rad D) 2π rad 3. A mass M hangs in equilibrium on a spring. M is made to oscillate about the equilibrium position by pulling it down 10 cm and releasing it. The time for M to travel back to the equilibrium position for the first time is 0.50 s. Which line, A to D, is correct for these oscillations?

Amplitude / cm Period / s A 10 1.0 B 10 2.0 C 20 2.0 D 20 1.0

4. Which one of the following statements is true when an object performs simple harmonic motion about a central point O? A) The acceleration is always away from O. B) The acceleration and velocity are always in opposite directions. C) The acceleration and the displacement from O are always in the same direction. D) The graph of acceleration against displacement is a straight line. 5. A body executes simple harmonic motion. Which one of the graphs, A to D, best shows the relationship between the kinetic energy, Ek, of the body and its distance from the centre of oscillation?

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6. The displacement (in mm) of the vibrating cone of a large loudspeaker can be represented by the equation x = 10 cos (50πt), where t is the time in s. Which line, A to D, in the table gives the amplitude and frequency of the vibrations.

amplitude/mm frequency/Hz

A 5 50/2π

B 10 25

C 10 25π

D 20 50/2π

7. 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 8. The tip of each prong of a tuning fork is vibrating at a frequency 320 Hz and amplitude of 0.5 mm. What is the speed of each tip when its displacement is zero? A) zero B) 0.32π mm s-1 C) 160π mm s-1 D) 320π mm s-1 9) A 5.0 kg object undergoes a simple harmonic motion. The potential energy U as a function of displacement x from its equilibrium position is as shown below. This particular system has a total energy of 0.40 J.

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What is the frequency of the simple harmonic motion? A) 2.5 Hz B) 16 Hz C) 50 Hz D) 250 Hz 10) Which one of the graphs, A to D, best shows how the displacement, x, of a damped oscillator that performs simple harmonic motion varies with time t?

11) Which one of the following statements about an oscillating mechanical system at resonance, when it oscillates with a constant amplitude, is NOT correct? A) The amplitude of oscillations depends on the amount of damping. B) The frequency of the applied force is the same as the natural frequency of oscillation of the system. C) The total energy of the system is constant D) The applied force prevents the amplitude from becoming too large.

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SIMPLE HARMONIC MOTION - 2012 1. ACJC P3Q5 5 (a) A rectangular block of wood of cross-section A and thickness t floats

horizontally in a water as shown in Figure 5.

The block floats when its lower face is at a depth d in the water of density ρ. The block experiences a force F on its lower surface as a result of immersion in the water.

(i) State the direction of the force F.

(ii) The pressure on the lower surface of the block due to the water is P.

Show that P is related to d, ρ and the acceleration of free fall g by the expression

P = dρg

[2] (iii) Using the expression in (ii), show that the force F is related to the

volume V of water displaced by the expression

F = Vρg

[2] (b) When the block is pushed down a further distance x into the water, show

that the expression of the resultant force FR is given by

FR = Axρg [2]

Direction of F is [1]

t

d

Area A

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(c) When the block is pushed downwards and then released, it undergoes damped simple harmonic motion.

(i) Using the result from (b), explain why the block is said to be

undergoing simple harmonic oscillation. [1]

(ii) Hence state the expression of the angular frequency of the motion of

the block in terms of A, ρ, g and m where m is the mass of the block.

(iii) Explain why the simple harmonic motion of the block is damped. [2]

(iv) Sketch a labeled graph showing the variation of displacement x with

time t for a time interval over three periods.

ω = m

gAρ

Expression for Angular frequency = [1]

x

t T 2T 3T

[2]

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(d) Surface water waves from a constant amplitude wave generator are

incident on the block. These causes forced oscillations in the motion of the block. The frequency of the wave generator is varied and resonance was observed at a particular frequency.

(i) Explain the terms given in italics with reference to the motion of the vibration of the block. [4]

(ii) Resonance occurs when the water waves incident on the block has a

speed of 0.90 m s−1 and wavelength 0.30 m.

1. Calculate the frequency of the water waves.

2. Given that the expression for the natural frequency of the

oscillating block is given by m

f28

2

1

π= where f is in Hz and m

in kg, calculate the mass of the block.

3. Describe and explain what happens to the amplitude of vertical oscillations of the block after the block has absorbed some water. [1]

forced oscillations: The water waves acts as an external periodic driving force continuously supplies energy to the damped system to compensate for the loss of energy due to the loss of energy which occurred during the oscillation of the system. The system will oscillate with a constant amplitude and with the frequency of the water waves. Resonance:

Frequency = 3 Hz [1]

mass = 0.0788 kg [1]

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2. DHSP2Q2 (a) A mass m, initially at rest on a smooth horizontal table, is attached by two identical light springs, each of spring constant k, to two fixed supports as shown below.

(i) When the mass is displaced an amplitude A, it oscillates in a simple harmonic motion with period T. The angular frequency w of the oscillation is

given by w =√2k m

. Show that the energy of the harmonic motion E is

E = 2π2mA2

T2 [1]

(ii) Deduce the energy of the system, in terms of T, A and m, if one of the springs is removed given that the amplitude remains the same. [2] (iii) The displacement-time graph of the system is given in Figure 1 below. Sketch, on the same axes, the displacement-time graph of the system if the oscillation is performed on a rough table surface. [1]

(b) (i) Explain what is meant by forced oscillation? [1] (ii) Explain how a singer can possibly break a glass by singing a particular note over time. [2]

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3. MJCP2Q3 A small delivery truck can be thought of as a box supported by four springs, one at each

wheel (the suspension of the truck). On a particular road, speed bumps are put on the road to slow down the traffic. After passing rapidly over one of these speed bumps, a delivery truck experiences rapid vertical oscillations.

Figure 3.1 Figure 3.2 shows a graph of acceleration, a, against displacement (from equilibrium), x, for the motion of the truck.

Figure 3.2

a Calculate the angular velocity ω of the truck. [2]

8.72 rads-1

b Calculate the shortest time taken t for the truck to oscillate from its lowest point to a

point 0.025 m below its equilibrium position. [3] 0.151s

c If the truck travels at a certain speed over the series of speed bumps, the vertical oscillations can be very large. Explain why this is so. [2]

Speed bumps

-0.10 0.10

a / m s-2

x / m

7.6

- 7.6

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4. (a) Explain what is meant by simple harmonic motion. [2] (b) Figure below shows a trolley of mass 1.0 kg tethered between two extended springs. The springs are identical and they obey Hooke’s Law.

Fig. 2.2 show the trolley displaced 0.10 m to the right. Both springs are still in tension.

(i) In Fig. 2.2, the resultant force on the trolley is 4.0 N to the left. Calculate the period of oscillation of the trolley. [3] 0.993 s (ii) After 100 oscillations, the amplitude is found to be halved. Calculate the percentage loss in energy. [2] (c) A modeling program produced the graph of Fig. 2.3. The program calculated the acceleration at time intervals of 0.2 s, and used these values to estimate the changes in velocity and displacement for the next time interval.

(i) Use the graph of Fig. 2.3 to estimate the period of the motion predicted by the model. [2] 0.93 s (ii) The graph obtained is not accurate. Suggest one method to improve the results.

[1]

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5. Question on car suspension system and resonance. The suspension system of a car can be considered to be a spring under compression with a shock absorber which damps the vertical oscillations of the car. The car is driven at a steady speed over a rough road on which the surface height varies sinusoidally. The shock absorber mechanism which normally damps vertical oscillations is not working. At a certain critical speed, the amplitude of the vertical oscillation of the car becomes very large. (i) State the phenomena observed. [1] (ii) On the axes of the figure below, sketch a graph to illustrate how, for a working shock absorber mechanism, the height of a car varies with time as it comes down from a road hump. Assume that the car tyres remain in contact with the ground throughout. [1]

(iii) The spring suspension system obeys Hooke’s law. Calculate the force constant, k, of the spring suspension system. The following data is given: Mass of passengers, m = 450 kg Mass of car and passengers, M = 2000 kg Difference in height of car when passengers alight, s = 0.10 m. [3] 44 100 N m-1

(iv) Hence, find the critical speed of the car when the amplitude of vertical oscillation is a maximum if the wavelength of the road corrugations is 20 m.

(the period of oscillation, T = 2π m k

[3]

14.9 m s-1

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(v) Sketch on the same axis appropriately labelled graphs to contrast how the amplitude of oscillation would vary at different speeds if the damping mechanism is 1. working 2. not working. [2] 6. (a) Distinguish between damped and forced oscillations. [2] (b) Indiana Jones is in the Temple of Doom, and his pathway is blocked by a huge blade that moves up and down with simple harmonic motion. The top edge of the blade moves all the way up to the roof, which is at a height of 2.20 m, and the bottom edge of the blade touches the ground every 4.0 s as shown in the figure below. The blade has a vertical width of 0.80 m.

If the only way Indiana can cross this obstacle is to roll under the blade, how much time does he have for rolling if he needs a space that is at least 0.55m? 2.2 s

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7. (i) Explain what is meant by the expression simple harmonic motion. [2] (ii) sketch a graph of acceleration against displacement to illustrate your explanation in (i). [1] (iii) A mass is placed on a flat horizontal platform and the platform is made to oscillate with simple harmonic motion in a vertical direction with an amplitude of 0.10 m. At a distance of 0.06 m from its equilibrium position, the speed of the platform is 0.30 m s-1. Determine (a) the period. [2] 1.7 s (b) the maximum speed of the platform. [1] 0.38 m s-1 (c) The amplitude is increased very slowly. Calculate the maximum amplitude of vibration so that the mass on the platform always remains in contact with it. 0.7 m

(iv) During an earthquake, the ground shakes at different frequencies. A certain frequency can result in large amplitude of vibration in a building within the earthquake area. (a) Give a brief explanation of this observation. Illustrate your answer with a suitable labelled graph. (b) Describe what will happen to the vibration of the building due to the earthquake when the building is equipped with additional devices such as friction dampers which utilize frictional forces to dissipate energy.

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8. A pendulum bob in a particular clock oscillates so that its displacement from a fixed point varies with time t as shown in the figure below.

The mass of the pendulum bob is 0.100 kg. Calculate (a) the angular frequency of the oscillations. [1] 6.28 rad s-1

(b) magnitude of the maximum velocity. [2] 0.754 m s-1

(c) the maximum kinetic energy of the system. [1] 2.82 x 10-2 J (d) Sketch the graph of velocity against displacement. [2] (e) Sketch on the same axes, the graphs of kinetic energy and potential energy against displacement. [3]

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9. (a) State two conditions for the motion of an oscillating body to be simple harmonic. [2] (b) A light spring is loaded with a mass of 200 g and made to execute vertical oscillations. The figure below shows the force-extension (N-mm) graph for the spring.

(i) With reference to the figure, explain why the oscillations are likely to be simple harmonic. [2] (ii) By considering the restoring force on the mass, show that the angular

frequency of the oscillations is given by ω = k

m , where k is the spring

constant and m is the loaded mass. [2]

(iii) Find the slope k of the graph. Hence, calculate the period of vertical oscillation of the 200 g mass. [2] 30 N m-1; 0.51 s (iv) Determine the equilibrium extension of the spring for the same suspended mass. [1] 0.065 m

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10. The figure below shows the variation with frequency f of the amplitude x, of the forced oscillations of a machine.

(a) State (i) what is meant by a forced oscillation. [1] (ii) the name of the effect illustrated in the figure. [1] (b) At any value of frequency, the oscillations of the machine are simple harmonic. Calculate for the machine vibrating at maximum amplitude, the maximum magnitudes of (i) Linear speed 1.2 m s-1

(ii) Linear acceleration 91 m s-2

(iii) Determine the time interval between the a maximum linear speed and the subsequent maximum linear acceleration. 0.021 s

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11. (a) For an object undergoing simple harmonic motion, state two quantities that are constant for the same motion. [2] (b) A body undergoes simple harmonic motion. The figures show how its velocity v and its kinetic energy K.E. varies with its displacement x respectively.

Deduce, from the numerical values given in the above figures, (i) the maximum potential energy, 1.0 x 10-3 J (ii) the mass of the body, 2.0 x 10-3 kg (iii) the displacement x, where the ratio of the body’s potential energy to its kinetic energy is 0.20. 4.1 mm

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12. A 75 kg diver is standing at the end of a diving board while it is vibrating up and down in simple harmonic motion, as shown in the figure. The diving board has an effective spring constant of k = 4100 N m-1, and the vertical distance between the highest and lowest points in the motion is 0.30 m.

(a) What is the amplitude of the motion? [1] 0.15 m (b) Starting when the diver is at the highest point, what is his speed one-quarter of a period later? [3] 1.1 m s-1

(c) Calculate the time required for the diver to make one complete motional cycle. [2] 0.85 s (d) Find the minimum frequency at which the diver just ceases to remain in contact with the platform. [2] 1.28 Hz

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13. 07 MJC P3/Q2

A 0.200 kg object is attached to a vertical spring which has a force constant of 4.00 N m-1. It is then pulled down by a distance of 8.00 cm and released. (a) Explain the type of motion that the object undergoes and state its defining equation. [2]

(b) Given that the period is given by T = 2πm k

, write down a numerical

equation for the displacement, x, of the object in terms of time, t. [2] x = – 0.08 cos (4.47t) (c) Hence or otherwise, calculate the speed, v, of the object when its displacement is 6.00 cm. [2] 0.237 m s-1 (d) Calculate the shortest time required for the object to move from x = +4.00 cm to −4.00 cm. [3] 0.234 s (e) Sketch the displacement-time graph over several periods of such an oscillation if the object were immersed in water. [1]

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14. 07 RI P2/Q5

(a)

Fig. 5 shows the variation with displacement x of the acceleration a of a 0.25 kg load attached to a spring. Use Fig. 5 to

(i) explain why the motion of the 0.25 kg load is simple harmonic, [2]

(ii) show that the period T of the load is 0.85 s, [3] (b) (i) Calculate the maximum kinetic energy of the load. [2] 0.011 J (ii) Use the space below to sketch a graph of kinetic energy against time for

two cycles of oscillation, given that the load was at maximum downward displacement at t = 0 s. Label the period T of the oscillation on the time axis. [2]

2.0 4.0 -4.0 -2.0 0

2.0

1.0

-1.0

-2.0

x / cm

a / m s−−−−2

Fig. 5

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15. 07 TJC P2/Q3 A brick hung on a vertical spring is pulled down from its equilibrium position and released. It travels up and then down to the position from which it was released in 0.75 s, travelling a total distance of 0.80 m. (a) Find the amplitude and frequency of the vertical oscillations. [2] 0.20 m 1.33 Hz

(b) Calculate the distance moved by the brick 0.12 s after it was released. [3] 0.09 m

(c) The top end of the vertical spring is now attached to a vibrator. Sketch a graph to show how the amplitude of the brick varies with frequency as the frequency of the vibrator is increased from 0 Hz to 3.0 Hz. [2]

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16. The set-up in Fig. 3 is used to investigate the phenomenon of resonance. There are two pendulums connected by a light spring. (1) Pendulum A is a compound pendulum consisting of a 1.00 m rod (mass not negligible but small), a movable mass mA that can slide along the rod and a paper card of negligible mass attached to the end of the rod that can be rotated about the axis of the rod. (2) Pendulum B can be treated as a simple pendulum consisting of an identical rod as Pendulum A but of length 1.50 m and a large cylindrical mass mB (mB >> mA) that can slide along the rod.

Pendulum B is set into oscillations with constant amplitude along the plane of the set-up and the amplitude of oscillation of pendulum A is measured. It is known that the natural frequency of pendulum A and B are respectively

fA = k x – 0.4 and fB =k

L

where k is the proportionality constant, x is measured from the point of suspension of pendulum A to the centre of mA, and L is measured from the point of suspension of pendulum B to the centre of the cylindrical mass mB. The motion of pendulum B is assumed to be unaffected by the oscillation of pendulum A.

(a) What is resonance? [2] (b) State the objectives of each of the following actions: (i) turning the paper card by 90º such that it is perpendicular to the plane of the set-up. [1]

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(ii) moving the mass mA along the rod in pendulum B. [1] (c) Suppose mass mA is moved to x = 0.500 m, and the length L of mB is varied. (i) Find the length L where resonance occurs. [1] 0.574 m (ii) Sketch on the same axes, using solid lines and with clear labels, the curves showing how the maximum amplitude of pendulum A depends on the length of pendulum B when 1. the plane of the paper card is along the plane of the set-up. 2. the plane of the paper card is perpendicular to the plane of the set-up. [2]

(iii) If mass mA is now at x = 0.750 m and the length L of mB is varied. Sketch on the same graph in (ii), using dotted line and with clear label, the curve showing how the maximum amplitude of pendulum A depends on the length of pendulum B. [1] (d) A stronger spring is now used in the set-up. Discuss the effect on the curves you have drawn in (c). [2]

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17. The interaction between water molecules can be modeled by simple harmonic motion. At 0 K, all molecules will be at their lowest energy and they will remain stationary at their equilibrium position. A rise in temperature results in the molecules vibrating about their equilibrium position. The total energy per molecule at temperature T, in Kelvins, is given by ETotal = 1.4 x 10-23 T J. (i) Calculate the total energy, ETotal, of individual water molecule at boiling point. 5.2 x 10-21 J (ii) At boiling point, the amplitude of vibration of the water molecules is 0.35 nm. On the same axis, sketch labelled graphs to show the variation of the energy with the displacement from equilibrium point for 1. the total energy, 2. the kinetic energy, and 3. the potential energy. (iii) Electromagnetic wave of certain frequency interacts with the water molecules by acting as a driving force. At this frequency, resonance occurs and water starts to boil. Find this frequency and name the region it occupies in the electromagnetic spectrum. (Mass of 1 water molecule = 3.0 x 10-26 kg) 2.7 x 1011 Hz [10]