1/A simple pendulum consists of an object suspended by a string. The object is a

Question

1/A simple pendulum consists of an object suspended by a string. The object is assumed to be a particle. The string, with its top end fixed, has negligible mass and does not stretch. In the absence of air friction, the system oscillates by swinging back and forth in a vertical plane. If the string is 2.15 m long and makes an initial angle of 32.5° with the vertical, calculate the speed of the particle at the following positions.
(a) at the lowest point in its trajectory

(b) when the angle is 15.0°

Explanation / Answer

here,

initial angle , theta1 = 32.5 degree

length of the string , L = 2.15 m

(a)

let the speed of particle at the lowest point in its trajectory be v

using conservation of energy

kinetic energy gained = change in potential energy

0.5 * m * v^2 = m * g * L * (1 - cos(32.5))

0.5 * v^2 = 9.8 * 2.15 * (1 - cos(32.5))

v = 2.57 m/s

the speed of particle at the lowest point in its trajectory is 2.57 m/s

(b)

let the speed of particle when angle theta2 = 15 deg be u

using conservation of energy

kinetic energy gained = change in potential energy

0.5 * m * u^2 = m * g * L * ((1 - cos(32.5)) - (1- cos(15)))

0.5 * u^2 = 9.8 * 2.15 * ((1 - cos(32.5)) - (1- cos(15)))

u = 2.27 m/s

the speed of particle when angle is 15 deg is 2.27 m/s

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