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Question
Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution and the initial position are indicated on the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P is ______.
Options
`x(t) = B sin ((2pit)/30)`.
`x(t) = B cos ((pit)/15)`.
`x(t) = B sin ((pit)/15 + pi/2)`.
`x(t) = B cos ((pit)/15 + pi/2)`.
Solution
Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution and the initial position are indicated on the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P is `underline(x(t) = B sin ((2pit)/30))`.
Explanation:
Suppose a particle P is moving uniformly on a circle of radius A with the angular speed. Q and R are the two feet of the perpendicular drawn from P on two diameters one along. the Y-axis and the other along the Y-axis.
 (a) |
 (b) |
Suppose the particle P is on the X-axis at t = 0. Radius OP makes an angle with the X-axis at time t, then x = A cosωt and y = A sinωt.
Here, x and v are the displacements of Q and R from the origin at time t, which are the displacement equations of SHM. It implies that although P is under uniform circular motion, Q and R are performing SHM about O with the same angular speed as that of P.
Let the angular velocity of the particle executing circular motion be ω and when it is at P make an angle θ as shown in the diagram.
As `sin θ = x/(OP) = x/B`,
Clearly, ω = ωt
x = B sin θ = B(sin ωt) = B sin `((pit)/15)`
x = B sin `((2pi)/30 t)`
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