For a body performing simple harmonic motion, the relation between velocity and displacement
v=ω√A2−x2Square 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=Asin‌ωtDifferentiate w.r.t. to
t‌=v=+ωA‌cos‌ω‌tAgain differentiate w.r.t. to
t,
‌=a=−ω2Asin‌ωtFrom Eq. (i),
a=−ω2xComparing from Eq.
y=mx+cIt 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),
‌=a=−ω2Asin‌ωtThe graph between acceleration and time will be a function of sine wave.
(d) From Eq.
(A),
‌v=ω√A2−ω2‌v2=ω2(A2−x2)‌x2=(−‌+A2)‌x=√A2−‌ Putting this value in Eq. (iv),
a=−ω2√A2−‌Square both side
a2‌=ω4(A2−‌)a2‌=ω4A2−v2ω2(‌)2‌=A2−‌(‌)2‌+(‌)2=A2When
ω≠1, the graph between acceleration
a as a function of velocity
v in SHM is an ellipse.