Question

State the expressions for the displacement, velocity and acceleration draw performing linear SHM, starting from the positive extreme position. Hence, their graphs with respect to time.

Answer 1

Starting from the positive extreme position, consider a particle executing linear SHM with amplitude A and time T = 2π/ω, where ω is the angular frequency. At every given time (t), its deviations from the mean position (x), velocity (v), and acceleration (a) are

x = A cos ωt = A cos `((2π)/Tt)`   ...`(∵ ω = (2π)/T)`

v = − ωA sin ωt = − ωA sin `((2π)/Tt)`

a = − ω²A cos ωt =  − ω²A cos `((2π)/Tt)`

Using these expressions, the values of x, v and a at the end of every quarter of a period, starting from t = 0, are tabulated below.

t	0	T/A	T/2	3T/4	T
ωt	0	π/2	π	3π/2	2π
x	A	0	− A	0	A
v	0	− ωA	0	ωA	0
a	− ω2A	0	ω2A	0	− ω2A

Using these values, we can plot graphs showing the variation of displacement, velocity and acceleration with time.

Variation of displacement, velocity and acceleration with time for a particle in SHM starting from the positive extremity