Graph of displacement against time 

Recall that the relation between displacement and time for a body
oscillating with s.h.m. is 



So, a graph of x against t has the same form as a graph of sine
of angle against angle. 





Graph of velocity against time 

Similarly, it has been shown that the relation between velocity and time is 



So, a graph of v against t has the same form but is shifted to the left 





Graph of acceleration against time 

In order to find the relation between acceleration and time we
simply take the basic equation which defines s.h.m. 



and substitute for x, giving 



So a graph showing how the acceleration of the body varies with time looks
like this 





If we remember that the slope of an x against t graph represents
velocity and the slope of a v against t graph represents
acceleration, then we can
more or less predict the shapes of the second two graphs by looking at the
slopes of various points on the previous one. 

For example 



These observations tell us that the v against t graph must start out at a
positive maximum, then curve down to zero, then reach a negative maximum (of
the same magnitude as the positive one) etc 

Of course, if you prefer a more mathematical sounding explanation... the
second equation is the derivative of the first and the third equation is the
derivative of the second. 



Graphs of Energy against Time 

Remember that an oscillation is a periodic conversion of energy
between kinetic and potential. 

If an oscillation is s.h.m., then the total energy possessed by the
oscillating body does not change with time. 

In practice, the total energy of most mechanical oscillations
decreases with time, usually because of air resistance or something
similar. 

An oscillation in which the total energy (and therefore the amplitude)
decreases with time is called a damped oscillation. 



Graph of Total Energy against Time 

As it was clearly stated above that the total energy is
constant, this one is fairly easy to predict! 





Graph of Kinetic Energy against Time 

The form of this graph can be deduced by starting from the velocity against
time graph and remembering that, in general, kinetic energy is given by 



So the KE possessed by a body oscillating with s.h.m. is 



Therefore the K.E. against time graph has the same form as a sin^{2}
graph, shown below. 

A copy of the v/t graph is added for comparison. 





Graph of Potential Energy against Time 

Knowing that the total energy is constant, allows us to deduce that the
potential energy graph must be have the same form as the kinetic energy
graph but inverted. 


