1. Two important examples of s.h.m.
© D Hoult 2010

2. The mass / spring oscillator

3. The mass / spring oscillator

5. net force acting on the mass = zero

9. A graph of force against extension for a spring is a straight line passing through the origin:
force a extension

10. A graph of force against extension for a spring is a straight line passing through the origin:
force a extension
The constant of proportionality, k is called the elastic constant of the spring (usually considered to be a positive number)

12. When the displacement is downwards, the net force is upwards and vice versa

13. So a better way to draw the graph would be

15. In this case, F represents the force with which the spring “pulls back” when it is given an extension x by some external agency

16. slope = - k

17. F = - k x

18. F = - k x
If the mass is released after having been given a displacement, x, then its acceleration will be
a =

19. F = - k x
If the mass is released after having been given a displacement, x, then its acceleration will be
- k x
a = m

20. F = - k x
If the mass is released after having been given a displacement, x, then its acceleration will be
- k x
a = m
So the motion is s.h.m. and the constant of proportionality between a and x has magnitude

21. F = - k x
If the mass is released after having been given a displacement, x, then its acceleration will be
- k x
a = m
So the motion is s.h.m. and the constant of proportionality between a and x has magnitude k m

22. F = - k x
If the mass is released after having been given a displacement, x, then its acceleration will be
- k x
a = m
So the motion is s.h.m. and the constant of proportionality between a and x has magnitude k m
which is equal to w2 in the “shm equation”

23. Remembering that 2p
T = w
we can suggest that the time period of a mass spring oscillator is given by

24. Remembering that 2p
T = w
we can suggest that the time period of a mass spring oscillator is given by m
T = 2p k

25. The simple pendulum

34. F =

35. F = mg sin q

36. F =

37. F = - mg sin q

38. a =

39. a = - g sin q

40. q =

41. x
q = L

42. If we make sure that q is a small angle

43. If we make sure that q is a small angle, sin q can be replaced by q

44. g
a = - x L