In simple harmonic motion, the total mechanical energy of the given system is \( E \). If the mass of the oscillating particle \( P \) is doubled, then the new energy of the system for the same amplitude is:
In a simple harmonic oscillator, the total mechanical energy (T.E.) is given by: \[ T.E. = \frac{1}{2}kA^2, \] where \( k \) is the spring constant and \( A \) is the amplitude of oscillation.
- Since the amplitude \( A \) remains the same, the total mechanical energy (T.E.) will also remain the same, as it depends only on \( k \) and \( A \), not on the mass \( m \) of the oscillating particle.
Thus, even if the mass of \( P \) is doubled, the total mechanical energy \( E \) will remain unchanged.
Answer: E
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Approach Solution -2
Step 1: Recall the expression for total energy in simple harmonic motion (SHM)
The total mechanical energy \(E\) in SHM is given by the formula:
\[
E = \frac{1}{2} k A^2
\]
where \(k\) is the spring constant and \(A\) is the amplitude of oscillation.
This energy is the sum of the maximum potential and kinetic energies in the system. It depends only on the amplitude and the stiffness (spring constant) of the system, not on the mass of the particle performing SHM.
Step 2: Consider what happens if the mass is doubled
When the mass of the oscillating particle changes, the natural frequency of oscillation changes, because the angular frequency is
\[
\omega = \sqrt{\frac{k}{m}}
\]
If the mass is doubled, the new angular frequency becomes:
\[
\omega' = \sqrt{\frac{k}{2m}} = \frac{\omega}{\sqrt{2}}
\]
This means the oscillations become slower, but the total mechanical energy at the same amplitude still depends on \(k\) and \(A\), not on \(m\).
Step 3: Verify by considering velocity and kinetic energy expressions
The maximum velocity in SHM is \(v_{\text{max}} = A \omega\).
So, total energy can also be written as:
\[
E = \frac{1}{2} m v_{\text{max}}^2 = \frac{1}{2} m (A^2 \omega^2)
\]
Substitute \(\omega^2 = \frac{k}{m}\):
\[
E = \frac{1}{2} m A^2 \frac{k}{m} = \frac{1}{2} k A^2
\]
Notice that the mass \(m\) cancels out, showing that the energy does not depend on \(m\). Therefore, even if the mass is doubled, the total mechanical energy remains unchanged for the same amplitude.
Step 4: Conceptual understanding
Increasing the mass decreases the angular frequency of oscillation, but since the spring constant and amplitude remain the same, the total energy stored in the oscillation stays constant. In SHM, energy depends only on how far the spring (or restoring force system) is displaced, not on how heavy the oscillating object is.
