Question

A simple pendulum of length 0.2 m has bob of mass 5 gm, it is pulled aside through an angle ${{60}^{0}}$from the vertical. A spherical body of mass 2.5 gm is placed at the lowest position of the bob. When the bob is released it strikes the spherical body and comes to rest. What is the velocity of the spherical body? (in m/s)
(g=9.8 $m/{{s}^{2}}$)
(A) 1.4
(B) 2.8
(C) 3.5
(D) 4.9

Answer 1

The length of the pendulum is 0.2 m and the mass of the bob is 5 g. When the pendulum is displaced from its mean position and then it moves to and fro in a simple harmonic motion. Now, the angle made with the vertical is ${{60}^{0}}$and the bob is taken to one of its extreme positions. Also, a body is placed at its mean position and the bob strikes the body. We can make use of law of conservation of energy and law of conservation of momentum.

Complete step by step answer:
Length of pendulum,$l=0.2m$
Mass of the bob,$m=5g$
Mass of the body $m'=2.5g$
Angle made with the vertical,$\theta ={{60}^{0}}$

We know potential energy of the bob of a pendulum is given by $mgh$. First of all, find the height of the pendulum at its extreme position.
From triangle $ABC$,
$\Rightarrow \cos 60=\dfrac{b}{0.2}$
$\Rightarrow \dfrac{1}{2}=\dfrac{b}{0.2}$
$\Rightarrow b=0.1$
So, the height of the pendulum is $0.2-0.1=0.1m$
So, when the bob moves and strikes the body its entire potential energy gets converted into kinetic energy of the Body.
$\Rightarrow mgh=\dfrac{m{{v}^{2}}}{2}$
$\Rightarrow 5\times 9.8\times 0.1=\dfrac{5{{v}^{2}}}{2}$
$\Rightarrow {{v}^{2}} = 1.96$
$\Rightarrow v=1.4m/s$
So, the velocity of the bob comes out to be 1.4 m/s. Now we can make use of the law of conservation of momentum.
$\Rightarrow 5\times 1.4=2.5v$
$\therefore v=2.8m/s$

So, the correct option is B.

Note: This is the easiest way to solve this problem. Momentum conservation is a universal law which holds until there are no external force acts on the system. Also, there is no loss of energy taking place in the system. All the units must be in standard SI convention.Had an external force been acting on the system, we would not be able to solve this problem using conservation of linear momentum law. Momentum is a vector quantity because it is the product of mass and velocity and velocity is a vector quantity.