HSC Science (Computer Science) 12th Board ExamMaharashtra State Board
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Solution - Show that S.H.M. is a Projection of U.C.M. on Any Diameter - HSC Science (Computer Science) 12th Board Exam - Physics

Question

Show that S.H.M. is a projection of  U.C.M. on any diameter

Solution

Suppose that particle P starts from the initial position with initial phase α (angle between radius OP and the x-axis at the time t = 0).
In time t, the angle between OP and x-axis is (ωt +α) as particle P moving with constant angular velocity (ω) as shown.

`cos(omegat+alpha)=x/a`

`thereforex=acos(omegat+alpha)`   .........(Equ.1)

This is the expression for displacement of particle M at time t

As velocity of the particle is the time rate of change of displacement then we have

`v=dx/dt=d/dt[acos(omegat+alpha)]`

`therefore v=-aomegasin(omegat+alpha)`   .............(Equ.2)

As acceleration of particle is the time rate of change of velocity, we have

`a=(dv)/dt=d/dt [-aomegasin(omegat+alpha)]`

`thereforea=-aomega^2cos(omegat+alpha)`

`therefore a=-omega^2x`

Hence, the projection of a uniform circular motion on a diameter of a circle is simple harmonic motion.

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APPEARS IN

Solution for question: Show that S.H.M. is a Projection of U.C.M. on Any Diameter concept: Projection of U.C.M.(Uniform Circular Motion) on Any Diameter. For the courses HSC Science (Computer Science), HSC Science (Electronics), HSC Science (General)
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