Simple Harmonic Motion

For a body performing simple harmonic motion, the relation between velocity and displacement
v=ω√A2−x2
Square both side
v2=ω2(A2−x2)
For ( ω=1 )
v2+x2=A2
(a) So, the graph between velocity and displacement will be a circle.
(b) We know that,
x=Asin‌ωt
Differentiate w.r.t. to t
‌

dx

dt

=v=+ωA‌cos‌ω‌t
Again differentiate w.r.t. to t,
‌

d2x

dt2

=a=−ω2Asin‌ωt
From Eq. (i),
a=−ω2x
Comparing from Eq. y=mx+c
It is a equation of straight line with slope, m=−ω2.
So, the graph between acceleration and displacement will be a straight line.
(c) From Eq. (iii),
‌

d2x

dt2

=a=−ω2Asin‌ωt
The graph between acceleration and time will be a function of sine wave.
(d) From Eq. (A),
‌v=ω√A2−ω2
‌v2=ω2(A2−x2)
‌x2=(−‌

v2

ω2

+A2)
‌x=√A2−‌

v2

ω2

Putting this value in Eq. (iv),
a=−ω2√A2−‌

v2

ω2

Square both side
a2‌=ω4(A2−‌

v2

ω2

)
a2‌=ω4A2−v2ω2
(‌

a

ω2

)2‌=A2−‌

v2

ω2

(‌

a

ω2

)2‌+(‌

v

ω

)2=A2
When ω≠1, the graph between acceleration a as a function of velocity v in SHM is an ellipse.