Question

Assuming the expression for displacement of a particle starting from extreme position, explain graphically the variation of velocity and acceleration w.r.t. time.

Answer 1

Displacement-time graph:
i.

At extreme position, α = π/2

Displacement, x = A cos ωt

Velocity time graph:
i.

At extreme position, α = π/2

Velocity of a particle is v = - Aω sin ωt

ii. Table:
Substituting ω = 2π/T  in above equation,

Time (t)	Phase (ωt)	Velocity (v)
0	0	0
T/4	π/2	-Aω
T/2	π	0
3T/4	3π/2	Aω
T	2π	0

Graph:

Acceleration-time graph:
i. At extreme position,α = π/2
  Acceleration of a particle is,
  a =-Aω2 cos ωt 

ii. Table:
  Substituting ω = 2π/T in above equation

Time (t)	Phase (ωt)	Velocity (v)
0	0	-Aω2
T/4	π/2	0
T/2	π	Aω2
3T/4	3π/2	0
T	2π	-Aω2

iii. Graph:

Conclusions:
i. Displacement, velocity and acceleration of S.H.M. are periodic functions of time.
ii. The displacement and acceleration curves are sine curves whereas velocity curve is
cosine curve (α = 0).

iii. The phase difference between displacement and acceleration is of π radian.
iv. The phase difference between displacement and velocity and velocity and acceleration
is of π/2 radian.
v. The displacement and acceleration is maximum at extreme position whereas velocity is
minimum at the same position.
vi. All curves repeat same path after phase of 2π radian.