Question

A body is executing S.H.M. At a displacement \(x\), its potential energy is \(9\;J\) and at a displacement \(y\), its potential energy is \(16\;J\). The potential energy at displacement \((x+y)\) is

• A. \(25\;J\)
• B. \(5\;J\)
• C. \(49\;J\)
• D. \(7\;J\)

Hint:

In S.H.M., \[ U\propto x^2 \] Always use the relation \[ U=\frac12 kx^2 \] to connect displacement and potential energy.

Answer 1

The Correct Option is C

Step 1: Recall the potential energy in SHM.
For a particle in simple harmonic motion, the potential energy at displacement $x$ from the mean position is \[ U = \frac{1}{2}kx^2 \] So $U$ is proportional to the square of the displacement.
Step 2: Use the proportionality with displacement.
Since $U \propto x^2$, we have $x \propto \sqrt{U}$. This lets us treat displacements like square roots of their energies.
Step 3: Translate the given data.
At displacement $x$, $U_x = 9\ J$, so $x \propto \sqrt{9} = 3$. At displacement $y$, $U_y = 16\ J$, so $y \propto \sqrt{16} = 4$.
Step 4: Add the displacements.
The combined displacement corresponds to \[ x + y \propto 3 + 4 = 7 \]
Step 5: Find the potential energy at $(x+y)$.
Since $U \propto (\text{displacement})^2$, \[ U_{x+y} \propto 7^2 = 49 \]
Step 6: State the answer.
Therefore the potential energy at displacement $(x+y)$ is $49\ J$, matching option (3). \[ \boxed{49\ J} \]