Q60 of 101 Page 132

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.

The force exerted by the particle on the sphere is


The distance travelled is


Given


The mass of the particle is given as m, fixed on a sphere of radius R, where the particle makes a radius with an angle of 300 with vertical.


Formula Used


Using the conservation of static and dynamic energy such as potential and centripetal energy, we have the conservation equation as



where


m is the mass of the object, v is the velocity, R is the radius of the circular path, r is the radius of the object, h is the height.


Explanation


(a) The mass of the particle when horizontal is given as




Hence, force exerted by the sphere is.


(b) The distance travelled by the particle in terms of radian/degree is calculated as


The change in potential energy due to the angle of is



Equating the kinetic and potential energy together we get the value of velocity as



And



The equation formed is







More from this chapter

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58

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°.



 59

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.

 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.

 62

Figure (8-E17) shows a smooth track which consists of a straight inclined part of length l joining smoothly with the circular part. A particle of mass m is projected up the incline from its bottom. (a) Find the minimum projection-speed u0 for which the particle reaches the top of the track. (b) Assuming that the projection-speed is 2u0 and that the block does not lose contact with the track before reaching its top, find the force acting on it when it reaches the top. (c) Assuming that the projection-speed is only slightly greater than u0, where will the block lose contact with the track?