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Question
Define angular S.H.M. and obtain its differential equation.
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Solution
Angular S.H.M. is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement and its direction is opposite to that of angular displacement.
- Consider a metallic disc hanging from rigid support, when twisted, it performs an oscillatory motion for which the restoring torque acting upon it, for angular displacement θ is,
τ ∝ -θ
∴ τ = -cθ ….(1) - The constant of proportionality (c) is the restoring torque per unit angular displacement.
- If I is the moment of inertia of the disc, the torque acting on the disc is given by,
τ = Iα .....(2)
Where α is the angular acceleration. - From equations (1) and (2),
Iα = -cθ
∴ `"I"("d"^2θ)/"dt"^2 + "c"θ = 0` .........`(∵ α = ("d"^2θ)/("dt"^2))`
This is the differential equation for angular S.H.M.
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