Question

A simple pendulum of length \[1m\] has a wooden bob of mass $1\,kg$ . It is struck by a bullet of mass ${10^{ - 2}}\,kg$ moving with a speed of $2 \times {10^2}\,m{s^{ - 1}}$ . The height to which the bob rises before swinging back is (Take $g = 10\,m{s^2}$ ).
A. $0.2m$
B. $0.6m$
C. $8m$
D. $1m$

Answer 1

Hint: To answer the question, we will first write all of the given values appropriately, then use the concept of conservation of momentum to get the combined speed of the bullet and the bob. The height will then be determined using the notion of energy conservation.

Complete step by step answer:
Let us first write all the given values accordingly, it is given to us in the question that, a simple pendulum with a length of \[1m\].Mass of it is given as \[1kg\].
Mass of the bullet is given as ${10^{ - 2}}kg = 0.01{\text{ }}kg$.
And speed with which it is moving is $2 \times {10^2}\,m{s^{ - 1}} = 200\,m{s^{ - 1}}$
Before swimming back, the bob rises to a certain height \[\left( {g = 10\,m{s^{ - 2}}} \right)\] .

After collision, the combined speed of the bullet and the bob equals \[V\]. Using the law of conservation of momentum
\[mv = (m + M)V \\
\Rightarrow (0.01)(200) = (0.01 + 1)V \\
\Rightarrow V = 1.98\,m{s^{ - 1}} \\ \]
Let us consider that $h = $ The height to which the pendulum's bob is lifted
According to conservation of energy;
Electric potential energy gained= Kinetic energy lost
\[(m + M)gh = (0.5)(m + M){V^2} \\
\Rightarrow V = \sqrt {2gh} \\
\Rightarrow 1.98 = \sqrt {2 \times 10 \times h} \\
\therefore h = {(1.98)^2} \times 2 \times 10 = 0.2\,m \]
Therefore, the height to which the bob rises before swinging back is \[0.2\,m\].

Hence, the correct option is A.

Note: It is important to realise that the tension force does not work. This is due to the fact that a force's work is defined as the dot product of the applied force and the body's displacement. The dot product of perpendicular vectors, on the other hand, is zero, and the tension and displacement of the bob are perpendicular to each other in this case. The speed of the bob can also be determined using the principles of mechanical energy conservation.