Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum. upwards (or to right) is positive and downwards (or to left) is negative.
(i) The phase space diagram for a ball thrown vertically up from ground is:
Text Solution
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(i)
(ii) In 1st case amplitude of SHM is a.
In 2nd case amplitude of SHM is 2a
Total energy = 
k(amplitude) 2
E 1 = 
k(2a) 2
E 2 = 
k(a) 2 
(iii) Linear momentum
P = mv
= 
⇒ P 2 + m 2 ꞷ 2 x 2 = m 2 ꞷ 2 A 2 represents a circle on P–x diagram with radius of circle R = A ( m
2 ꞷ 2 = 1) ꞷ of spring mass system remains constant and equal to 
Amplitude of oscillation inside liquid will decrease due to viscous force So radius of circular arcs will decrease as position change Correctly shown in option B
P = mv
= 
⇒ P 2 + m 2 ꞷ 2 x 2 = m 2 ꞷ 2 A 2
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