A simple pendulum consists of a small bob of mass suspended from a light
PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics
Problem
A simple pendulum consists of a small bob of mass m suspended from a light, inextensible string of length L. For small angular displacements θ (measured in radians), the motion of the bob can be approximated as simple harmonic.
Consider the bob when it is at an angle θ from the vertical. The arc-length displacement from equilibrium is s=Lθ.
Answer the questions below about the restoring force acting on the bob for small θ.
(a) Starting from the component of the gravitational force that acts tangentially to the bob’s path, show that for small angles the tangential restoring force can be written Ft≈−mgθ.
(b) Express this force in the form Ft=−ks by substituting s=Lθ, and identify an effective force constant k in terms of m, g, and L.
(c) For a pendulum of length 0.80m and bob mass 0.25kg, calculate the magnitude of the restoring force when the bob is 2.0cm along the arc from its lowest point. Verify that the small-angle approximation is valid for this displacement.
(d) Explain how the dependence of k on L would affect the period of small oscillations if the length of the pendulum were doubled, even though the bob’s mass remains the same.