[total mechanical energy r...] A small body attached at the end of an inextensible string completes a vertical circle, then its

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Aptitude

TNPSC G4

10th CBSE Maths

A small body attached at the end of an inextensible string completes a vertical circle, then its

angular velocity remains constant

angular momentum remains constant

total mechanical energy remains constant

linear momentum remains constant

Explanation:

Whenever a body is released from height, it travels vertically downward towards the surface of earth. This is due to the force of gravitational attraction exterted on body by the earth. The acceleration produced by this force is called acceleration due to gravity and is denoted by ‘g’. Value of ‘g’ on the surface of earth is taken to the 9.8 m/s2 and it is same for all the bodies. It means all bodies (whether an iron ball or a piece of paper), when dropped (u=0) from same height should fall with same rapidity and should take same time to reach the earth. Our daily observation is contrary to this concept. We find that iron ball falls more rapidlly than piece of paper. This is due to the presence of air which offers different resistance to them. In the absence of air both would have taken same time to reach the surface of earth.
A particle of mass m is attached to a light and inextensible string. The other end of the string is fixed at O and the particle moves in a vertical circle of radius r equal to the length of the string as shown in the figure.
Forces Acting on the Particle
Consider the particle when it is at the point P and the string makes an angle θ with vertical. Forces acting on the particle are:
T = tension in the string along its length, and
mg = weight of the particle vertically downward.
Hence, net radial force on the particle is FR = T - mg cos θ
=> T - mg cos θ = mv2/R
=> T = mv2/R + mg cos θ
Since speed of the particle decreases with height, hence tension is maximum at the bottom, where cos θ = 1 (as θ = 0).
=> Tmax = mv2/R + mg; Tmin = mv'2/R - mg (at the top)
Here, v' = speed of the particle at the top.

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