Q. 585.0( 2 Votes )

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°.



Answer :

As the , the maximum height of the bob is equivalent to the initial height.


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 is


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° is


Given


The pendulum has a mass of “m” attached to length “l” which hits the peg hanging at “x” situated at an angle of .


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 l is the length of the object.


Explanation



a) When the height of the bob is less than the peg than the total potential energy of the bob at point A is equal to the potential energy of the bob at point B, as shown in the diagram



And


The kinetic energy at both places is equal to zero.


As the , the maximum height of the bob is equivalent to the initial height.


b)



When the particle is released at an angle of and x=L/2, hence, the path of the bob travelling will slack at point C making a projectile motion.






Hence, the distance after slack at point D is



c) The velocity of the bob at point D as shown in the figure 58.b is



The conservation of the force in the bob is given as






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