Question

The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 2m, what is the speed with which the bob arrives at the lowermost point? Assume that 10% of its energy is dissipated against air resistance. (Take $g = 10m{s^{ - 2}}$).
A. $4m{s^{ - 1}}$
B. $6m{s^{ - 1}}$
C. $8m{s^{ - 1}}$
D. $10m{s^{ - 1}}$

Answer 1

The given problem indicates a situation for a simple pendulum. The speed of the pendulum can be found with the help of conservation of energy during motion. So here we apply this method to solve this problem.

Complete step by step answer:
The given data in above problem is
Length of pendulum ($l$)= 2m
Dissipated energy against air resistance= 10% of total energy
Here, we want to find the speed of bob when it arrives at the lowermost point. So we will use the concept of conservation of energy to solve this problem.
Let’s consider that the bob is released from point A and at point B (Lowermost point), we have to find the velocity of the bob. The situation is shown in the figure.

Total energy of system= Kinetic energy + Potential energy
We will apply energy conservation from point A to point B.
Potential energy at point A= mgh
Kinetic energy at point A= 0 (Because velocity at point A=0)
Potential energy at point B= 0 (Taking point B as reference level)
Kinetic energy at point B= $\dfrac{1}{2}m{v^2}$
Where m= Mass of bob
g= Gravitational acceleration (9.8 m/s2)
h= Height from reference level
v= Speed at lowermost point

Now we apply energy conservation. Which implies that at lower most point B, whole potential energy is converted into kinetic energy. But according to question, 10% of the energy of the bob is dissipated against air resistance. So only 90% energy converted into kinetic energy at point B.
Total energy at point A = Total energy at point B
$\dfrac{1}{2}m{v^2} = \dfrac{{90}}{{100}}mgh \\\ \Rightarrow {v^2} = 2 \times 0.9 \times g \times h \\\ \Rightarrow v = \sqrt {2 \times 0.9 \times 9.8 \times 2} \\\ \therefore v = 6m{s^{ - 1}} \\\$
So the value of velocity of bob at the lowest point will be 6 m/sec.

Hence, the correct answer is option (B).

Note: The motion of a simple pendulum is considered as simple harmonic motion in which the mean position of motion exists at the lowermost point. Where bob has maximum value of velocity.