Show that simple harmonic motion may be regarded as the projection of uniform circular motion along a diameter of the circle.hence derive an expression for the displacement of a particle in shm?
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abiramiragu
Secondary SchoolPhysics 5+3 pts
Show that simple harmonic motion may be regarded as the projection of uniform circular motion along a diameter of the circle.hence derive an expression for the velocity of a particle in shm?
Report by SAKSHISANKHE002 20.03.2017
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abiramiragu
Abiramiragu · Virtuoso
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kvnmurty
Kvnmurty The Sage
In simple harmonic motion, the equation of motion is
d²x/dt² = - ω² x ,
ω= angular frequency, x = displacement from the mean position, t = time
Solution is : x = A Cos ωt for x=A, t=0 initial position.
When a particle is moving along a circular path of radius r, the position of the projection along the diameter (x axis) is the x coordinate. The speed v is constant. So the angular speed = ω = v/r. Angle made by the radial vector with x axis = Ф. As ω is constant, Ф = ωt
x = r cosФ = r cos Фt
comparing the equations of motion of SHM and this equation, we say that the projection does a SHM with amplitude = r = radius.
v = dx/dt = - Aω sin ωt = -ω √(A²-x²)