Q59 of 101 Page 132

A particle slides on the surface of a fixed smooth sphere starting from the topmost point. Find the angle rotated by the radius through the particle, when it leaves contact with the sphere.

The angle rotated by the radius through the particle, when it leaves contact with the sphere is


Explanation


The particle moving portrays a centripetal force of and a force of, now equating the forces we get the equation as


Calculating the value of velocity we get the value as




Hence, the real height is



Change in potential energy due to height is



Now placing the kinetic and potential energy together we get the change in angle as







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A heavy particle is suspended by a 15 m long string. It is given a horizontal velocity of m/s. (a) Find the angle made by the string with the upward vertical, when it becomes slack. (b) Find the speed of the particle at this instant. (c) Find the maximum height reach by the particle over the point of suspension. Take g = 10 m/s2.

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A simple pendulum of length L having a bob of mass m i deflected from its rest position by an angle θ and released (figure 8-E 16). The string hits a peg which is fixed at a distance x below the point of suspension and the bob starts going in a circle centred at the peg.

(a) Assuming that initially the bob has a height less than the peg, show that the maximum height reached by the bob equals its initial height.


(b) If the pendulum is released with θ = 90o and x = L/2 find the maximum height reached by the bob above its lowest position before the string becomes slack.


(c) Find the minimum value of x/L for which the bob goes in a complete circle about the peg when the pendulum is released from θ = 90°.



 60

A particle of mass m is kept on a fixed, smooth sphere of radius R at a position, where the radius through the particle makes an angle of 300 with the vertical. The particle is released from this position. (a) What is the force exerted by the sphere on the particle just after the release? (b) Find the distance travelled by the particle before it leaves contact with the sphere.

 61

A particle of mass m is kept on the top of a smooth 8phere of radius R. It is given a sharp impulse which imparts it a horizontal speed u. (a) Find the normal force between the sphere and the particle just after the impulse. (b) What should be the minimum value of u for which the particle does not slip on the sphere? (c) Assuming the velocity u to be half the mininium calculated in part, (d) find the angle made by the radius through the particle with the vertical when it leaves the sphere.