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How can a pendulum be used to trace out a sinusoidal curve?

How can a pendulum be used to trace out a sinusoidal curve?

Grade:upto college level

1 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
A simple pendulum is consisting of a particle suspended by alight inextensible cord. When pulled to one side of its equilibrium position and released, the pendulum swings in a vertical plane under the influence of gravity. The motion is periodic and oscillatory. Below figure shows a pendulum of length L and particle mass m.
235-2462_1.PNG
At the instant shown, the cord makes an angle θ with the vertical. The forces acting on mass m are the weight mg and the tension T in the cord. The motion will be along an arc of the circle with radius L, and so we choose axes tangent to the circle and along the radius. The weight mg is resolved into a radial component of magnitude mg cos θ and a tangential component of magnitude mg sin θ. The tangential component is the restoring force acting on mass m tending to return it to the equilibrium position. So the restoring force is
Fx = -mg sin θ.
Negative sign indicates that Fx is opposite to the direction of increasing x and increasing θ.
The restoring force is not proportional to the angular displacement θ. It is proportional to sin θ. That is why a pendulum can be used to trace out a sinusoidal curve.

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