Figures 14.25 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.
a)
Given: The radius of the circle is 3 cm and the period of revolution is 2 s .
At time t = 0 the radius of vector OP makes an angle π 2 with the positive x axis.
ϕ = π 2
The equation of SHM for the x projection of OP, at time t is given by the displacement equation,
x = A cos [ 2 π t T + ϕ ] (1)
Where, the time period is T , the amplitude is A and the phase angle is ϕ .
By substituting the values in equation (1), we get
x = 3 cos ( 2 π t 2 + π 2 ) = − 3 sin ( 2 π t 2 ) x = − 3 sin ( π t ) cm
Thus, the equation of SHM is x = − 3 sin ( π t ) cm .
b)
Given: The radius of the circle is 2 m and the period of revolution is 4 s .
When the value of amplitude and time period is changed, then
T = 4 s A = 2 m
At time t = 0 ; OP makes an angle of π in the anticlockwise direction.
The phase angle is given by,
ϕ = 180 °
By substituting the given values in equation (1), we get
x = 2 cos [ 2 π t 4 + ϕ ] x = 2 cos [ π t 2 + ϕ ] m
Thus, the equation of SHM is x = 2 cos [ π t 2 + ϕ ] m .
a)
Given: The radius of the circle is
At time
The equation of SHM for the
Where, the time period is
By substituting the values in equation (1), we get
Thus, the equation of SHM is
b)
Given: The radius of the circle is
When the value of amplitude and time period is changed, then
At time
The phase angle is given by,
By substituting the given values in equation (1), we get
Thus, the equation of SHM is
y - projection of the radius vector of rotating particle P is 
