Question

A simple pendulum of length 1m is released from horizontal position. If 20% its initial energy is lost due to air resistance in reaching the mean position, then the speed of the bob of the pendulum at mean position is (Acceleration due to gravity = 10 m s\textsuperscript{-2})

• A. 2 m s\textsuperscript{-1}
• B. 4 m s\textsuperscript{-1}
• C. 3 m s\textsuperscript{-1}
• D. 5 m s\textsuperscript{-1}

Hint:

\textbf{Conservation of Energy:} In the absence of non-conservative forces (like air resistance), mechanical energy (Potential Energy + Kinetic Energy) is conserved. \textbf{Pendulum Energy:} When a pendulum is released from rest, its initial energy is purely potential energy (mgh). At the lowest point (mean position), its energy is purely kinetic energy (\( \frac{1}{2} \)mv\textsuperscript{2}). \textbf{Energy Loss:} If energy is lost (e.g., due to air resistance), the final mechanical energy will be less than the initial mechanical energy.

Answer 1

The Correct Option is B

Step 1: Calculate the initial potential energy.
Length of the pendulum (L) = 1 m
The pendulum is released from the horizontal position, meaning its initial height (h\textsubscript{initial}) is equal to its length, L.
h\textsubscript{initial} = 1 m Acceleration due to gravity (g) = 10 m s\textsuperscript{-2} Let the mass of the bob be 'm'. Initial potential energy (PE\textsubscript{initial}) = mgh\textsubscript{initial} = m \( \times \) 10 \( \times \) 1 = 10m Joules. Step 2: Calculate the energy remaining at the mean position.
At the mean position, the height (h\textsubscript{final}) is 0. All the remaining potential energy is converted into kinetic energy. 20% of the initial energy is lost due to air resistance. So, 80% of the initial energy remains. Energy remaining = 0.80 \( \times \) PE\textsubscript{initial} = 0.80 \( \times \) 10m = 8m Joules. Step 3: Relate the remaining energy to the kinetic energy at the mean position.
At the mean position, the remaining energy is entirely kinetic energy (KE\textsubscript{final}). KE\textsubscript{final} = \( \frac{1}{2} \)mv\textsuperscript{2} Where 'v' is the speed of the bob at the mean position. So, \( \frac{1}{2} \)mv\textsuperscript{2} = 8m Step 4: Solve for the speed 'v'.
Cancel 'm' from both sides:
\( \frac{1}{2} \)v\textsuperscript{2} = 8
v\textsuperscript{2} = 16
v = \( \sqrt{16} \)
v = 4 m s\textsuperscript{-1}
Step 5: Select the correct option.
The speed of the bob at the mean position is 4 m s\textsuperscript{-1}, which matches option (2).