First slide

Law of conservation of momentum and it's applications

Question

A U-shaped wire has a rough semicircular bending between A and B as shown in the figure. A bead of mass m moving with uniform speed v through the wire enters the semicircular bend at A and leaves at B with velocity v/2 after time T. The average force, exerted by the bead on the part AB of the wire is

Moderate

A

$\frac{m v}{2 T}$
B

$\frac{3 m v}{2 T}$
C

$\frac{3 m v}{T}$
D

none of these

Solution

The momentum of the bead at A is ${\vec{p}}_{i} = - m v \hat{i}$
The momentum of the bead at B ${\vec{p}}_{f} = (m v / 2) \hat{i}$
Therefore, the magnitude of the change in momentum between A and B is
$Δ p = |{\vec{p}}_{f} - {\vec{p}}_{i}| = (3 / 2) m v$
Average force exerted by the bead on the wire is
$F_{a v} = \frac{Δ p}{Δ t} = \frac{3 m v}{2 T}$

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