Question

Figures 14.29 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.


Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

Answer 1

see the figure .
If particle moves from P to P' in time interval t . then , angle moved by position vector = radius vector .
∅ = wt = 2πt/T
T = 2sec
∅ = πt rad --------(1)
Displacement covered by particle
ON = OP'sin∅
Also ON = -x(t) [ negative direction of x-axis ]
Hence, -x(t) = OP'sin∅
Op' = 3 cm
So, x(t) = -3sin∅
x(t) = -3sinπt [ from eqn(1)



If the particle moves from P to P' in the time interval t .
Then angular displacement ( ∅) = wt
= 2π/T × t
Here T = 4 sec
So, ∅ = πt/2
Displacement ( ON) = OP'cos∅
But ON = -x(t)
-x(t) = OP'cos∅
x(t) = -2 cos∅ [ OP' = 2 m
x(t) = -2cos(πt/2)