A simple pendulum of length L having a bob of mass m is deflected from its rest position by an angle θ and released. 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.If the pendulum is released with θ=90∘ and x=L/2 find the maximum height reached by the bob above its lowest position before the string becomes slackf the pendulum is released with θ=90∘ and x=L/2 find the maximum height reached by the bob above its lowest position before the string becomes slack.
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∘.
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∘.
Energy before release = mgL
At point of slack => mg cosø = 2mv^2/L
v^2 = Lg cosø/2
Energy is conserved
So – mgL = m Lg cosø /4 + mg (L/2 + Lcosø/2)
4gL = 3 Lg cosø + 2Lg
2Lg = 3Lg cosø
cosø = 2/3
height reached = L/2 + L cosø /2 = 5L/6
Energy is conserved-
L-x = radius
when string slacks – mv’^2/(L-x) = mg
1/2mv^2 = mg(L-x)/2
mgL = 1/2mv’^2 + 2mg(L-x)
gL = 1/2 g(L-x) +2 g(L-x)
L = (L-x)/2 + 4(L-x)/2
L = 5L-5x/2
2L = 5L – 5x
-3L=-5x
x/L = 3/5 = 0.6