Calculate the velocity of the bob of a simple pendulum at its mean position if it is able to rise a vertical height of 10 cm. (Take g = 10 m/s2)

1. Break Down the Problem

To find the velocity of the bob of a simple pendulum at its mean position, we can use the principle of conservation of mechanical energy. The potential energy at the highest point (when the bob rises to a height of 10 cm) will be converted to kinetic energy at the mean position.

2. Relevant Concepts

• Gravitational Potential Energy (PE): $PE = mgh$
• Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$
• Conservation of Energy: $PE_{initial} = KE_{final}$

3. Analysis and Detail

1. Calculate the gravitational potential energy at the height of 10 cm (which is 0.1 m): $PE = mgh = mg \times 0.1$

2. Calculate the kinetic energy at the mean position: $KE = \frac{1}{2}mv^2$

3. Setting the potential energy equal to the kinetic energy: $mg \times 0.1 = \frac{1}{2}mv^2$

4. We can cancel the mass $m$ from both sides (assuming it is not zero): $g \times 0.1 = \frac{1}{2}v^2$

5. Rearranging the equation to solve for $v$: $v^2 = 2g \times 0.1$ $v = \sqrt{2g \times 0.1}$

6. Plugging in $g = 10 \, \text{m/s}^2$: $v = \sqrt{2 \times 10 \times 0.1} = \sqrt{2}$

4. Verify and Summarize

Calculating the final value: $v = \sqrt{2} \approx 1.41 \, \text{m/s}$

Final Answer

The velocity of the bob of the simple pendulum at its mean position is approximately $1.41 \, \text{m/s}$.