Question

The mass of bob of a simple pendulum of length L is m. If the bob is projected horizontally from the mean position with a velocity root 8gl, the tension in the string in horizontal position is

Answer 1

see attached figure,

Let mass of the bob of a simple pendulum of length L is m. if the bob is projected horizontally from the keen position with velocity $v=\sqrt{8gl}$.

when T is the tension in string in horizontal direction.

then, T = mv'²/l, ........(1)

where v' is the of simple pendulum when it is placed horizontally.

we know, from conservation of energy theorem,

$K.E_{mean}+P.E_{mean}=K.E_{horizontal}+P.E_{horizontal}$

or, 1/2 mv² = 1/2 mv'² + mgl

or, v² = v'² + 2gl

or, 8gl - 2gl = v'² [ as v = √8gl ]

or, v' = √6gl .........(2)

from equations (1) and (2),

now, T = mv'²/l = m(6gl)/l = 6mg

hence, tension in the string in the horizontal direction is 6mg