SHM - springs
Q1
(a) A body is moving with simple harmonic motion. State two conditions that must be satisfied concerning the acceleration of the body.
For a body to be moving with simple harmonic motion acceleration must be:
- proportional to displacement
![](../../../../graphics/symbols/nuclides/ticksmall.png)
- in the opposite direction to displacement, OR always acting towards a fixed point, or towards the centre of oscillation
![](../../../../graphics/symbols/nuclides/ticksmall.png)
(2 marks)
(b) A mass is suspended from a vertical spring and the system is allowed to come to rest. When the mass is now pulled down a distance of 76 mm and released, the time taken for 25 oscillations is 23 s.
Calculate
frequency = 25/23 = 1.09
f = 1.1 Hz ![](../../../../graphics/symbols/nuclides/ticksmall.png)
(ii) the maximum acceleration of the mass,
a = (2πf)2A
a = (2π × 1.09)2 × 76 × 10−3 ![](../../../../graphics/symbols/nuclides/ticksmall.png)
a = 3.6 m s−2 ![](../../../../graphics/symbols/nuclides/ticksmall.png)
(iii) the displacement of the mass from its rest position 0.60 s after being released. State the direction of this displacement.
x = A cos (2πft)
x = 76 × 10−3 cos (2π × 1.09 × 0.60) ![](../../../../graphics/symbols/nuclides/ticksmall.png)
x = − 4.3 × 10−2 m
x = 43 mm
direction: above equilibrium position or upwards ![](../../../../graphics/symbols/nuclides/ticksmall.png)
(6 marks)
(c)The diagram below shows qualitatively how the velocity of the mass varies with time over the first two cycles after release.
![](../q1.png)
(i) Using the axes below, sketch a graph to show qualitatively how the displacement of the mass varies with time during the same time interval.
![](../a1b.png)
correct shape, i.e. cos curve ![](../../../../graphics/symbols/nuclides/ticksmall.png)
correct phase i.e. −(cos) ![](../../../../graphics/symbols/nuclides/ticksmall.png)
(ii) Using the axes in below sketch a graph to show qualitatively how the potential energy of the mass-spring system varies with time during the same time interval.
graph to show:
two cycles per oscillation ![](../../../../graphics/symbols/nuclides/ticksmall.png)
correct shape (even if phase is wrong) ![](../../../../graphics/symbols/nuclides/ticksmall.png)
correct starting point ![](../../../../graphics/symbols/nuclides/ticksmall.png)
(4 marks)
(Total 12 marks)