A small block of mass m which is initially at rest, slides down without friction

Question

A small block of mass m which is initially at rest, slides down without friction through height h0 and then collides with a uniform rod of the same mass m and length L and sticks to the end of the rod. The rod pivots about point O to angle ? before momentarily stopping. After collision the motion of the rod – block system can be considered simple harmonic motion. In terms of L, h0, and g (acceleration due to gravity):

a)
Find an expression for the angular speed ? of the rod – block system immediately after collision.

b)
Using conservation of energy, find an expression for the maximum height h the block reaches after collision.

c)
Find an expression for the maximum angle ?

Explanation / Answer

a) velocity of block just before collison = v

mv2 / 2 = mgh

v = 2gh0  

by conservimg angular momentum ,

mvr = I

m X L X 2gh0   = (mL2 + mL2 /3 )

mL2gh0   =( 4mL2 /3 )

= 3(2gh0 )/4L

b ) I2 / 2 = mgh

(4mL2 /3 ) X 9 X 2gh0 / 2 x 16 L2    = mgh + mg(h/2)

3mgh0 / 4 = 3mgh/2

h = h0 / 2

c) h = L (1-cos)

h0 / 2L = 1 - cos

cos = (2L - h0 ) / 2L

= cos-1 ((2L - h0 ) / 2L)

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