A pendulum consists of a stone with mass m swinging on a string of length L and

Question

A pendulum consists of a stone with mass m swinging on a string of length L and negligible mass. The stone has a speed of v0 when it passes its lowest point. (a) Write an expression for the speed of the stone when the string is at some angle ? with the vertical. (b) What is the greatest angle with the vertical that the string will reach during the stone's motion? (c) If the potential energy of the pendulum-Earth system is taken to be zero at the stone's lowest point, what is the total mechanical energy of the system? State all your answers in terms of the given variables and g. I will give 5 stars for the correct answer.

Explanation / Answer

we will use conservation of energy for all parts

a) when the string is ? deg to the vertical, its height above its lowest point is

L-L cos ? = L(1-cos ? ) , where L is the length of the string

at the bottom of the swing, the stone has 1/2 m v0^2 kinetic energy

when it is ? deg from vertical, the total energy is 1/2 m V^2 + m gh

where V is the speed at the new height

therefore

1/2 m (v0)^2 = 1/2 m V^2 + m g h

v = ((v0)^2 -gh)1/2

b) if the stone converts all its KE to PE, its height above the lowest level will be given by

1/2 m (v0)^2 = m gh or h =(v0)^2/ 2*g

recall from above that height above lowest point = L(1-cos(theta))

so here(v0)^2/ 2*g = L(1-cos(theta))

theta = cos-1(1 - (v0)^2/ 2*L*g)

c) total mechanical energy = PE + KE

if PE =0 at the bottom, then the total mechanical energy = KE at bottom = 1/2 x m x (v0)^2

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