In SHM restoring force is `F = -kx`, where k is force constant, x is displacement and a is amplitude of motion, then total energy depends upon

Options

A. k, a and m
B. k, x,m
C. k, a
D. k,x

Answer

To solve the problem of determining what the total energy in Simple Harmonic Motion (SHM) depends upon, we will follow these steps:

### Step 1: Understand the Restoring Force
The restoring force in SHM is given by the equation:
\[ F = -kx \]
where \( k \) is the force constant and \( x \) is the displacement from the equilibrium position.

### Step 2: Identify Kinetic Energy
The kinetic energy (KE) of a particle in SHM can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 3: Express Velocity in SHM
For a particle performing SHM, the velocity can be expressed as:
\[ v = \omega \sqrt{A^2 - x^2} \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of motion.

### Step 4: Substitute Velocity into Kinetic Energy
Substituting the expression for \( v \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m (\omega \sqrt{A^2 - x^2})^2 \]
\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]

### Step 5: Identify Potential Energy
The potential energy (PE) in SHM is given by:
\[ PE = \frac{1}{2} kx^2 \]

### Step 6: Relate \( k \) to \( \omega \)
We can relate \( k \) to \( \omega \) using the formula:
\[ \omega^2 = \frac{k}{m} \]
Thus, we can substitute \( k \) as \( k = m\omega^2 \).

### Step 7: Substitute Potential Energy
Substituting \( k \) into the potential energy formula gives:
\[ PE = \frac{1}{2} (m\omega^2)x^2 \]

### Step 8: Calculate Total Energy
The total energy \( E \) in SHM is the sum of kinetic and potential energy:
\[ E = KE + PE \]
Substituting the expressions for KE and PE:
\[ E = \frac{1}{2} m \omega^2 (A^2 - x^2) + \frac{1}{2} (m\omega^2)x^2 \]
\[ E = \frac{1}{2} m \omega^2 A^2 \]

### Step 9: Final Expression for Total Energy
Thus, the total energy in SHM can be expressed as:
\[ E = \frac{1}{2} m \omega^2 A^2 \]

### Step 10: Identify Dependencies
From the final expression, we can see that the total energy depends on:
- Mass \( m \)
- Angular frequency \( \omega \)
- Amplitude \( A \)

### Step 11: Replace \( \omega \) with \( k \)
Since \( \omega^2 = \frac{k}{m} \), we can also express total energy in terms of \( k \):
\[ E = \frac{1}{2} \left(\frac{k}{m}\right) m A^2 = \frac{1}{2} k A^2 \]

### Conclusion
Thus, the total energy in SHM depends on the force constant \( k \) and the amplitude \( A \).

---