Question

A block of mass M is suspended by a massless wire of length L and, after a completely inelastic collision with a bullet of mass m, swings up to a maximum height y < L. Find: (a) the speed v of the block with the bullet in it immediately after impact; (b) the tension force T in the wire immediately after impact; (c) the initial horizontal velocity vo of the bullet. Explain pendulum tension please.

Answer 1

here,

The height the bob rises is related to theta as
h=L*(1-cos(theta))
the energy required to raise the bob is
(M+m)*g*h

this is related to the speed of the bob after collision as

0.5*(M+m)*vc^2=(M+m)*g*h

and it is related to v0 using conservatin of momentum

m*v0=(M+m)*vc

combine it all together now

v0=(M+m)*vc/m

vc = sqrt(2*g*y)

v0 = (M+m)*sqrt(2*g*y)/m

a)

theta = arccos((L - y)/L)

using conservatin of momentum

m*v0=(M+m)*vc

sqrt(2*g*L*(1-cos(theta))) = vc

vc = sqrt(2*g*L*(1-cos(theta)))

vc = sqrt(2*g*L*(1- ( 1 - y/L))

vc = sqrt( 2*g*y)

the speed vc of the block with the bullet in it immediately after impact is vc = sqrt(2*g*y )

b)

the tension in the string after the collison , T = (m + M ) * g + (m + M) * vc^2 /L

T = ( m + M) * ( g + *(2*g*y)^2 /L)

the tension in the string is T = ( m + M) * ( g + *(2*g*y)^2 /L)

c)

v0 = (M+m)*sqrt(2*g*y)/m