### SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION

1. Periodic motion:-
Any motion that repeats itself along the same path in equal intervals of time is called ‘periodic motion’.
Examples:-
1. The motion of a pendulum of wall clock.

2. Back and forth motion of the balance wheel of a wrist watch.
3. The motion of strings in musical instruments like violin, guitar, veena etc.,.
4. Motion of a mass attached to a spring.
5. To and fro motion of molecules of air as a sound wave passes by and
6. The invisible motion of atoms in a solid.
Note:-
Periodic motion is also known as “harmonic motion”.

2. Oscillatory or vibratory motion:-
If a particle in a periodic motion moves back and forth over the same path its motion is said to be oscillatory or vibratory motion.

3. Simple Harmonic Motion:-(SHM)
The to and fro motion of a particle about a mean position on a fixed path such that the acceleration of the particle is always directed towards the mean position and is directly proportional to the displacement of the particle from its mean position is called a “simple harmonic motion”.

4. Examples of Simple Harmonic Motion:-
1. The oscillations of a simple pendulum are simple harmonic.
2. The vibrations of the prongs of a tuning fork are simple harmonic.
3. The oscillations of a swing are harmonic.
4. The vertical oscillations of a loaded spiral spring are simple harmonic.
5. The oscillations of a liquid in a U-tube are simple harmonic.

5. Characteristics of a Simple Harmonic Motion:-
Simple Harmonic Motion is characterized by
1. A constant time period (T) or a constant frequency, 2. An amplitude (A) and
3. a constant mechanical energy which is the sum of potential energy and kinetic energy at every point in the path of oscillation.
6. Under which conditions a body will be in SHM
The motion should be
1. Periodic.
2. It should move on either sides about a fixed point.
3. The acceleration should be proportional to its displacement.
4. The acceleration and displacement should be opposite to one another.

7. Describe an experiment to determine acceleration due to gravity by using a Simple Pendulum? 1. Note the length of the simple pendulum (l).
2. Pull the bob aside slightly and release it.
3. It begins to execute SHM.
4. Count 20 oscillations and switch off the stop watch exactly at the end of 20th oscillation.

5. Note the time ‘t’ for 20 oscillations.

6. After two trials take average ‘ta’ of the ‘t’ values and hence determine the time period 7. Repeat the experiment for different lengths and tabulate the results.

 Sl.no l cm Time for 20 oscillations (sec) Average time (sec) ta Time period T2 (sec2) 1st trial 2nd trial 3rd trial 1 2 3 4 5 6 30 40 50 60 70 80

8. By substituting value in the formula We can determine the value of g at a given place.

8. What happens to the time period of simple pendulum in a moving train
When train is moving with uniform velocity there is no acceleration to pendulum. But, when the acceleration of the train increases or decreases it affects the time period of the pendulum.

9. when does a periodic motion become oscillatory motion
If a particle in a periodic motion moves back and forth over the same path then its motion becomes oscillatory or vibratory motion.

10. Give examples for oscillatory motion observed in your daily life.
Pendulum of a wall clock, oscillations of a swing.

Note:-
Time period of the pendulum is by Where,
l= length of the pendulum
g=acceleration due to gravity.
11. Seconds pendulum:-
A pendulum whose time period is 2 seconds is known as seconds pendulum.

II. Problems:-

1. Find the time period of a simple pendulum whose length is 100 cm (g=9.8m/s2).
Given,
l=100cm=1m.
g=9.8m/s2
T=?
We know that, T=2.006(≈2 sec)
Time period of the pendulum=2 seconds.

2. Time period of a simple pendulum on the moon is 5 sec. If the length is 105 cm. Find its acceleration due to gravity on the moon.
Given,
Time period of the pendulum on the moon =T=5 sec
Length of the pendulum =l=105 cm 3. Find the length of the simple pendulum whose time period is 1.2 sec (g=9.8 m/s2).
Given 4. In the simple pendulum experiment the value of l/T2 is found to be 0.248 m/s2 . Find the value of ‘g’ at that place.
Given, g=4×69.14×0.0354
g=9.790224
g=9.8 ms2.
Acceleration due to gravity ‘g’ at that place = 9.8m/s2.

III. Fill in the blanks:-

1. An example of an oscillatory motion observed in daily life is         (a)
a) Motion of string in a guitar.
b)motion of a bicycle along a curved path.
c)motion of a train along a curved path.
d)earth revolving around the sun.

2. The motion in which the particle repeats its motion along the same path is oscillatory motion.

3. Any motion that repeats itself along the same path in equal intervals of time is called periodic motion.

4. If a particle in a periodic motion moves back and forth along the same path is said to have oscillatory motion.

5. The fixed position mid way between the displacements on left and right side of a vibrating particle is called (b)
a)stationary position. b)mean position.
c)phase difference. d)radius vector.

6. In case of a balance wheel of a watch the forces responsible for oscillatory motion are (a)
a)Restoring force. b)inertial force.
c)coherent force. d)Mechanical force.

7. In case of a simple pendulum the force responsible for oscillatory motion is (c).
a)restoring force. b)mechanical force.
c)gravitational force. d)repulsive force.

8. If the acceleration of a moving object is always directed towards a fixed point on its path and is proportional to its displacement from its fixed point, its motion is said to be simple harmonic motion.

9. When the pendulum is oscillating in which position the velocity is maximum (b)
a)At the extreme position.
b)At the mean position.
c)At the neighbour hood of the extreme position.
d)None of the above.

10. Relation between acceleration ‘a’ of a particle and its displacement ‘x’ is given by aα-x.

11. If the frequency of the vibrating particle is ‘υ’ and its time period is ‘T’ then ‘υ’ = .
12. One of the following characterizes simple harmonic motion (a)
a)amplitude. b)phase.
c)mean position d)All the above.

13. The foot of a the perpendicular drawn to the diameter from a particle under going uniform circular motion makes simple harmonic motion.

14. The time period of a simple pendulum is directly proportional to square root of length of the pendulum.

15. If ‘l’ is the length of the simple pendulum and ‘T’ is the time period of oscillation then (a). 16. If ‘l’ is the length of the simple pendulum and ‘T’ is its time period and ‘g’ is acceleration due to gravity at that place then g 17. The period of oscillation of a seconds pendulum is 2 seconds.

18. Periodic motion is also known as harmonic motion.
19. if a particle in a periodic motion moves back and forth over the same path the motion is called oscillatory motion.

20. A body executing oscillatory motion comes to rest at mean position.

21. In the case of a simple pendulum it is the gravitational force that makes it oscillate.

22. At the extreme position a particle executing simple harmonic motion will have maximum acceleration.

23. If the acceleration is proportional to the displacement of a body from a fixed point then the motion of the body is called SHM.

24. One complete to and fro motion is called oscillation (or) vibration.

25. When a particle is performing SHM its potential energy will be minimum at mean position of the particle.

26. When a particle is performing SHM its kinetic energy will be maximum at mean position of the particle.

27. When a particle is performing SHM its potential energy will be maximum at extreme position of the particle and kinetic energy will be minimum at extreme position.

28. The time taken to complete one oscillation by a vibrating body is called time period.

29. A body in a oscillatory motion does so because of continuous action of a force whose magnitude varies but its direction remains unchanged.

30. To and fro vibration of an excited tuning fork is an example of Simple Harmonic Motion.

31. A component force due to gravity acts on the centre of the mass of the water contained in a U-shaped glass tube when it is displaced from its equilibrium level.
32. g= .

33. Sound waves are able to propagate because of elastic restoring forces of air medium.

34. A S.H.M. is associated with every wave motion.

35. Any motion that repeats itself along the same path in equal intervals of time is called periodic motion.

36. In a S.H.M. the acceleration of a particle is directly proportional to displacement of a particle from its mean position.

37. The displacement of a particle in periodic motion can be mathematically expressed in terms of sine or cosine function and so it is known as harmonic motion.
38. All periodic motion need not be oscillatory motion.

39. The position of rest of the pendulum that is midway between its displacement is called its mean position or equilibrium position.

40. A particle in a uniform circular motion executes periodic motion but not oscillatory motion.

41. The waves formed by the oscillation of magnetic and electric field vectors are called electro magnetic waves.

42. Examples of electro magnetic waves are radio waves, micro waves, light waves etc.,.

43. sound waves are able to propagate in air because of elastic restoring forces of air medium.

44. The maximum displacement of a vibrating body from its mean position is called amplitude.

45. visible light is an example of electro magnetic waves.

Q: Are all Vibratory motions SHM? Justify your answer.

Ans. All vibratory motions are Simple Harmonic Motion. All vibratory motions satisfy the conditions for SHM.
Consider spring – mass system . if mass attached to the spring is displaced, the mass starts oscillating about the mean position. Here the conditions for Simple Harmonic Motion are satisfied