Question

A child is sitting on a swing which performs S.H.M. It has minimum and maximum heights from the ground as $0.75\text{ m}$ and $2\text{ m}$ respectively. Its maximum speed will be [$g = 10\text{ m/s}^2$]

• A. $1.25\text{ m/s}$
• B. $12.5\text{ m/s}$
• C. $5\text{ m/s}$
• D. $25\text{ m/s}$

Hint:

The maximum velocity calculation for an oscillating swing or pendulum is identical to finding the final velocity of an object falling freely from a height of $\Delta h$. Using the standard kinematic shortcut $v = \sqrt{2g\Delta h}$ bypasses S.H.M. notation completely.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The swing behaves as a simple pendulum executing Simple Harmonic Motion (S.H.M.). The problem gives the lowest position (minimum height $h_{\min} = 0.75\text{ m}$) and highest position (maximum height $h_{\max} = 2\text{ m}$) of the child from the ground level. We need to determine the maximum speed of the swing.

Step 2: Key Formula or Approach:
By the

Law of Conservation of Linear Mechanical Energy, the loss in gravitational potential energy when moving from the highest extreme position to the lowest mean position must equal the total gain in kinetic energy at the mean position: $$\Delta E_k = \Delta U \implies \frac{1}{2}mv_{\max}^2 = mg(h_{\max} - h_{\min})$$ We can cancel the mass $m$ from both sides and isolate $v_{\max}$: $$v_{\max} = \sqrt{2g(h_{\max} - h_{\min})}$$

Step 3: Detailed Explanation:
Identify and isolate the parameters from the text: Maximum height, $h_{\max} = 2\text{ m}$ Minimum height, $h_{\min} = 0.75\text{ m}$ Acceleration due to gravity, $g = 10\text{ m/s}^2$
Calculate the net vertical displacement height change ($\Delta h$): $$\Delta h = h_{\max} - h_{\min} = 2 - 0.75 = 1.25\text{ m}$$ Now substitute these values into the derived velocity conservation formula: $$v_{\max} = \sqrt{2 \times 10 \times 1.25}$$ $$v_{\max} = \sqrt{20 \times 1.25} = \sqrt{25}$$ $$v_{\max} = 5\text{ m/s}$$ The maximum speed achieved at the lowest point of the swing path is $5\text{ m/s}$.

Step 4: Final Answer:
The maximum speed is $5\text{ m/s}$, which matches option (C).