A pendulum consist of a particle P of mass m , attached to one end of a model st

Question

A pendulum consist of a particle P of mass m, attached to one end of a model string of length l = 1.85 m. The other end of the string is attached to a fixed point O and the particle moves in a vertical circle about the fixed point as shown.

The angle made by the string with the vertical is measured anticlockwise and we assume that the only forces acting on the particle are its weight W and the tension force T due the string. The particle starts at the lowest point with an initial horizontal velocity of ?(7gl/2) in the anticlockwise direction.

Calculate the speed of the particle, in m s-1 when the tension becomes zero, giving your answer to 3 decimal places.

Please show all working clearly.

Explanation / Answer

equation of force alon the string

T-mgcos a -mv^2/l =0 where a =theta

when T=0

gcosa =- v^2 /l,lcos a=-v^2/g

from energy equation at T=0

1/2 mv^2 + mg( l-lcos a)= 1/2 m V^2

1/2 v^2 +g(l+v^2/g)=1/2 V^2

3v^2+2gl =V^2

3v^2 +2gl= 7gl/2

3v^2=3gl/2

v=sqrt gl/2=sqrt (9.81*1.85/2)=3.012 m/s

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