Q53 of 101 Page 132

The bob of a pendulum at rest is given a sharp hit to impart a horizontal velocity where l is the length of the pendulum. Find the tension in the string when (a) the string is horizontal, (b) the bob is at its highest point and (c) the string makes an angle of 60o with the upward vertical.

The tension at the string, horizontal is T= 8mg


The tension at the string, highest point is T’ = 5mg


The tension at the string, at a 60o from the normal is T’’= 6.5mg


Given


The horizontal velocity of the pendulum after hit is , and “l” is the length of the pendulum.


Formula Used


The formula for the total energy in terms of kinetic and potential energy is given as



&



where


The is the total energy in terms of kinetic and potential energy, m is the mass of the object, g is the acceleration in terms of gravity and r is the length of the rod/thread attached with the bob, m is the mass of the bob, is the velocity of the bob.


Explanation


(a). Let the velocity of the bob at horizontal plane (B) be and the velocity when the bob reaches the lowest point (A) of the bob circle is taken as. Hence, the relationship between kinetic and potential energy be






Therefore, making the tension in the horizontal position as, placing



(b). When the bob reaches at the top of the circle that is opposite to point A the velocity here is taken as . Hence the velocity is





Now taking in the tension when the bob is at height C, the bob also experience the downward force of making the velocity as



(c) The velocity at point D which is at from the normal is taken as, making the velocity as





Therefore, the tension is taken as



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