1. [2082]

1. Show that the total energy of the particles executing SHM is constant. [3]
2. Draw a graph showing the variation of potential energy of particle in SHM with displacement. [1]
3. The time period of simple pendulum on the surface of earth is T. What will be its time period on the Moon’s surface? [1]

2. [2081 GIE ‘A’] Figure shows the schematic diagram of a motion of a simple pendulum. Show that the motion of a simple pendulum is simple harmonic motion (SHM). [2]

3. [2081 GIE ‘B’] Derive an expression for total energy of a particle executing simple harmonic motion. [3]
4. [2081 ‘B/C’] Write an expression for total energy of a particle executing SHM. Also draw a graph showing the variation of K.E. and P.E. of it. [2]
5. [2081 ‘D’]

1. Define simple harmonic motion. [1]
2. Obtain an expression for frequency of oscillation of vertical mass spring system. [2]

6. [2080 GIE ‘A’]

1. Define simple harmonic motion. [1]
2. Derive an expression for total mechanical energy of a particle executing simple harmonic motion. [3]
3. Show graphically, how the kinetic energy, potential energy and total energy of the particle vary with displacement from its equilibrium position. [1]

7. [2080 ‘P’] For a simple pendulum, show that the acceleration is directly proportional to the displacement of bob from its mean position. [2]
8. [2080 ‘R’]

1. The displacement (y) of a particle executing simple harmonic motion is [latex]y = rsin\omega t[/latex], where r is the amplitude of vibration.

1. Define amplitude. [1]
2. Calculate the acceleration of the motion. [2]

2. If a uniform spring is cut in half, what is the force constant of each half? How would the frequency of half spring differ from that using the same mass and entire spring? [1+1]

9. [2079 GIE ‘A’] What is simple harmonic motion? Derive an expression for time period in angular simple harmonic motion. [3]
10. [2079 GIE ‘B’] Define S.H.M. and write its equation. [1+1]
11. 11. [2079 ‘O’] Simple harmonic motion is defined from periodic functions like sine or cosine functions.

1. State the basic equation of motion for a body executing simple harmonic motion. [1]
2. Find expression for velocity and acceleration of a particle describing SHM. [2]
3. The tip of tuning fork goes through 550 complete vibrations in 1 sec. Find the angular frequency and time period of the motion. [1 + 1 = 2] Ans: 3454 rad/s, 0.0018 sec

12. 12. [2079 ‘V’]

1. What is meant by Simple Harmonic Motion? [1]
2. Show that motion of a simple pendulum is simple harmonic and hence calculate its time period. [3]
3. On what factors does the time period of simple pendulum depend? [1]

Old Course
SHORT ANSWER QUESTION

13. [2074 ‘B’] If the length of second’s pendulum is increased further by 200 percent, will it lose or gain time?      
14. [2073 ‘S’] What do you understand by a second’s pendulum? If it is taken to moon, will it gain or lose time? Why?  
15. [2072 ‘S’] A simple harmonic motion is represented, in usual notion by y = a sin[latex](\omega t+\phi)[/latex]. Find its acceleration.  

LONG ANSWER QUESTION

16. [2076 ‘B’] What is simple harmonic motion? Calculate the total energy of a particle executing simple harmonic motion.
17. [2076 ‘A’] What is simple harmonic oscillator? Obtain an expression for its total energy, and time period.
18. [2073 ‘S’] Show that the small oscillations of a mass loaded spring suspended vertically are simple harmonic. Deduce expression for its time period.
19. [2073 ‘D’] Define simple harmonic motion. Deduce a relation for total energy of a simple harmonic oscillator.
20. [2072 ‘S’] What is a simple pendulum? Show that motion of the bob of a simple pendulum is simple harmonic. Obtain an expression for its frequency.
21. [2072 ‘C’] What are characteristics of simple harmonic motion? Show that motion of vertical mass-spring system is simple harmonic and hence derive formula for its time period.

1. [2081 GIE ‘A’] A simple pendulum 2m long swings with an amplitude of 0.1m. Calculate the velocity of the pendulum at its lowest point and its acceleration at extreme ends. [3] Ans: 0.22 m/s and 0.5 m/s²
2. [2081 GIE ‘B’] On an average a human heart is focused to beat 85 times in a minute. Calculate its frequency and time period.                                    [2] Ans: 1.42 Hz, 0.7 sec
3. [2081 B/C] One end of a light spring having spring constant 18 N/m is attached to a rigid support. A mass of 0.15 kg is suspended from other end of the spring. A student pulls down it such that extension of spring increases by 4 cm. The student now releases it, as a result, the mass performs SHM.

1. Define simple harmonic motion.                                                       [1]
2. Calculate the maximum acceleration and time period of SHM. [2] Ans: 4.8 m/sec², 0.57 sec

4. [2081 ‘D’] A simple pendulum of effective length 4 m swings with an amplitude of 0.2 m. Compute the velocity of pendulum at its lowest point. [g = 9.8 [latex]ms^{-2}[/latex]]       [2] Ans: 0.32 m/sec
5. [2080 ‘P’] A bob of mass 8 kg performs SHM of amplitude 30 cm. The restoring force is 60 N. Calculate:

1. Time period
2. The maximum acceleration
3. Kinetic energy when displacement is 12 cm                           [3] Ans: (i) 1.256 sec (ii) 7.5 m/sec² (iii) 7.56 J

6. [2079 GIE ‘A’] A particle of mass 0.25 Kg oscillates with a period of 2 sec. If its greatest displacement is 0.4 m, what is its maximum kinetic energy?     [2] Ans: 0.2 J
7. [2079 GIE ‘B’] Calculate the period of oscillation of a simple pendulum of length 1.8 m with a bob of mass 2.2 kg. If the bob of this pendulum is pulled aside a horizontal distance of 20 cm and released, what will be the K.E. of the bob at the lowest point of swing.      [3] Ans: 0.245 J
8. [2076 ‘GIE’ ‘A’] A particle of mass 0.3 kg vibrates with a period of 2 seconds. If its amplitude is 0.5 m, what is its maximum kinetic energy? Ans: 0.37 J
9. [2076 ‘GIE’ ‘B’] A simple pendulum 4 m long swings with an amplitude 0.2m. Compute i. velocity of the pendulum at its lowest point, ii. its acceleration at the end of its path. Ans: 0.32 m/sec, 0.5 m/s²
10. [2076 ‘C’] The velocity of a particle executing simple harmonic motion is 16 cms^-1 at a distance of 8 cm from the mean position and 8 cms^-1 at a distance of 12 cm from the mean position. Calculate the amplitude of the motion. Ans: 0.1308 m
11. [2075 ‘B’] The position of a certain object in S.H.M. is given as x = 0.05 cos(290t + 2.5); where x is in meter and t is in sec. What are the amplitude, period and initial phase angle for this motion? Ans: 0.05 m, 0.022 sec, 2.5 rad.
12. [2074 ‘A’] A body of mass 0.1 kg is undergoing simple harmonic motion of amplitude 1 m and period 0.2 second. If the oscillation is produced by a spring, what will be maximum value of the force and the force constant of the spring? Ans: 98.6 N, 98.6 N/m
13. [2072 ‘E’] A body of mass 2 kg is suspended from a spring of negligible mass and is found to stretch the spring 0.1 m. What is its force constant and the time period? Ans: 200 N/m, 0.63 sec
14. A glider with mass m = 2.00 kg sits on a frictionless horizontal air track, connected to a spring with force constant k = 5.00 N/m. You pull the glider, stretching the spring 0.100 m and then releases it with no initial velocity. The glider begins to move back toward its equilibrium position (x = 0). What is its velocity when x = 0.080 m? Ans: 0.1 m/s
15. A simple pendulum 5 m long swings with an amplitude 25 cm. Find the velocity of the pendulum at its lowest point and the acceleration at the end of its path. Ans: 0.35 m/s, 0.5 m/s²
16. A body is vibrating with simple harmonic motion of amplitude 15 cm and frequency 4 Hz. Calculate the maximum value of acceleration and velocity. Ans: 3.77 m/s, 94.8 m/s²
17. A small mass of 0.2 kg is attached to one end of helical spring and produces an extension of 15 mm. The mass is now set into vertical oscillation of amplitude 10 mm. What is:
    i.    the period of oscillation?
    ii.   the maximum kinetic energy of the mass?
    iii.  the potential energy of the spring when the mass is 5mm below the centre of oscillation? (g = 9.8 [latex]ms^{-2}[/latex]). Ans: (i) 0.246 s (ii) [latex]6.5\times 10^{-3}[/latex] J (iii) [latex]1.67\times 10^{-3}[/latex] J
18. A simple pendulum has a period of 4.2 second, when the pendulum is shortened by 1m the period is 3.7 second. From these measurements, calculate the acceleration of free fall and the original length of the pendulum. Ans: 10 m/s², 4.5 m
19. A second pendulum is taken to the moon. If the time period on the surface of the moon is 4.90 seconds, what will be the acceleration due to gravity of the moon? Take acceleration due to gravity of the moon to be [latex]\frac{1}{6^{th}}[/latex] that of the earth. Ans: 1.63 m/s²
20. Calculate the period of oscillation of a simple pendulum of length 1.8 m with a bob of mass 2.2 kg. If the bob of this pendulum is pulled aside a horizontal distance of 20 cm and released. What will be the values of (i) the K.E. and (ii) the velocity of the bob at the lowest point of the swing? Ans: 0.243 J, [latex]0.47\ ms^{-1}[/latex]
21. The displacement y of a mass vibrating with simple harmonic motion is given by [latex]y = 20 sin10\pi t[/latex], where, y is in millimeter and t is in second. What is:
    i.    amplitude (ii) the period (iii) the velocity at t = 0 Ans: (i) [latex]2\times 10^{-3}[/latex] m (ii) 0.2 s (iii) 0.628 m/s
22. A particle of mass 0.3 kg vibrates with a period of 2 seconds. If its amplitude is 0.5 m. What is its maximum kinetic energy? Ans: [latex]3.7\times 10^{-2}[/latex] J
23. A small mass rests on a horizontal platform which vibrates in simple harmonic motion with a period of 0.25s. Find the maximum amplitude of the motion which will allow the mass to remain in contact with the platform throughout the motion. Ans: 0.0158 m
24. A simple pendulum 4m long swings with an amplitude of 0.2 m.
    a.   Compute the velocity of the pendulum at its lowest point.             b.   Compute its acceleration at the end of its path. Ans: 0.316 m/s, 0.499 m/s²