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.

Explanation / Answer

a) Let h is the height above its mean position when it makes angle with vertical.

so, h = L*(1 - cos(theta))

Use Work-energy throeem

Workdone by gravity = change in kinetic energy

-m*g*h = (1/2)*m*(v(theta)^2 - vo^2)

-2*g*h = v(theta)^2 - vo^2

v(theta) = sqrt(vo^2 - 2*g*h)

= sqrt(vo^2 - 2*g*L*(1 - cos(theta) ) <<<<----Answwer

b) when theta is maximum, v(theta) = 0

so,

0 = sqrt(vo^2 - 2*g*L*(1 - cos(theta) )

vo^2 - 2*g*L*(1 - cos(theta_max) ) = 0

1 - cos(theta_max) = vo^2/(2*g*L)

cos(theta_max) = 1 - vo^2/(2*g*L)

theta_max = cos^-1( 1 - vo^2/(2*g*L) ) <<<<----Answwer

c) Total mechaincal energy at every point = (1/2)*m*vo^2

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