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A small block of mass m held compressed against a spring as shown in the figure. The spring has force constant of k such that mg/k = 3m. After releasing the block it collides a spheres of same mass which is suspended from a massless rod of length 3m. The rod can freely rotated in a vertical circle, about its top point. Considering the coefficient of restitution 0.5, find the minimum compression in the spring so that sphere can complete full circle.

A small block of mass m held compressed against a spring as shown in the figure. The spring has force constant of k such that mg/k = 3m. After releasing the block it collides a spheres of same mass which is suspended from a massless rod of length 3m. The rod can freely rotated in a vertical circle, about its top point. Considering the coefficient of restitution 0.5, find the minimum compression in the spring so that sphere can complete full circle.

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Grade:12th pass

2 Answers

daksh
13 Points
9 years ago
you must specify the distance between end of spring and the rod
 
Aakash
110 Points
9 years ago
Let Velocity of sphere Just After collision be = v1
Let Velocity of sphere Just Before collision be = u1 = 0
Let Velocity of Block Just After collision be = v2
Let Velocity of Block Just Before collision be = u2
To complete full circle rod just require to reach top point with zero velocity at top
By conservation of energy
mg2L = (½)mv12
=> v1 = 2(gL)1/2     --------(1)
e = (v1+v2)/u2 = ½
v2 = (u2/2) –v1     ----- (2)
Conservation of momentum
mu2 = mv1 – mv2
u2 = v1 - v2
Using (2)
u2 = v1 – (u2/2) + v1
v1 = (3u2)/4
u2 = (4v1)/3
Using (1)
u2 = (8(gL)1/2)/3      -------- (3)
For minimum compression P.E in spring get completely converted to K.E
(Kx2)/2 = (½)mu22
(mgx2)/6 = (mu22)/2   --------- Using K=mg/3
Using (3)
(mgx2)/6 = (64mgL))/18
Solving we get
x = 8(L/3)1/2
L=3 (given)
x=8m
Minimum compression is 8m 

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