A pendulum consists of a 1.7-kg bob attached to a light 2.5-m-long string. While

Question

A pendulum consists of a 1.7-kg bob attached to a light 2.5-m-long string. While hanging at rest with the string vertical, the bob is struck a sharp horizontal blow, giving it a horizontal velocity of 4.7 m/s. At the instant the string makes an angle of 30° with the vertical, calculate the following:
(a) the speed
(m/s)

(b) the gravitational potential energy (relative to its value is at the lowest point)
(J)

(c) the tension in the string?
(N)

(d) What is the angle of the string with the vertical when the bob reaches its greatest height?
(degrees)

Explanation / Answer

Data: Mass of the bob, m = 1.7 kg Length of the string, L = 2.5 m Velocity at the lowest point, v = 4.7 m/s Angle at the given position, = 30 deg ( with vertical ) Solution: Formula for vertical height from the lowest point, h = L ( 1 - cos ) (a) Potential + kinetic energy at the given position =                                                 Kinetic energy at lowest position m g h + (1/2) m v'^2 = (1/2) m v^2 g h + (1/2) v'^2 = (1/2) v^2 2 g h + v'^2 = v^2 2 g L ( 1 - cos ) + v'^2 = v^2 2 * 9.8 * 2.5 * ( 1- cos 30 ) + v'^2 = 4.7^2 v'^2 = 22.09 - 6.56 v' = 3.94 m/s Ans: Velocity at the given position, v' = 3.94 m/s (b) Gravitational potential energy, PE = m g h                                                    = m g L ( 1 - cos )                                                    = 1.7 * 9.8 * 2.5 * ( 1 - cos 30 )                                                    = 5.58 J Ans: Potential energy at the given position, PE = 5.58 J (c) Tension, T = ( mv'^2 / L ) + mg cos                  = ( 1.7 * 3.94^2 / 2.5 ) + 1.7 * 9.8 * cos 30                  = 10.56 + 14.43                  = 24.99 N Ans: Tension in the string, T = 24.99 N (d) PE at greatest height = KE at the lowest point m g L ( 1 - cos' ) = (1/2) m v^2 g L ( 1 - cos' ) = (1/2) v^2 9.8 * 2.5 * ( 1 - cos ' ) = 0.5 * 4.7^2 1 - cos ' = 0.451 cos ' = 0.549 ' = 56.69 deg Ans: Angle at the greateast height, ' = 56.69 deg                                    

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