A particle of mass `m` moving along x-axis has a potential energy `U(x)=a+bx^2` where a and b are positive constant. It will execute simple harmonic motion with a frequency determined by the value of

Options

A. b alone
B. b and a alone
C. b and m alone
D. b,a and m alone

Explanation

To solve the problem, we need to analyze the potential energy function given and derive the frequency of the simple harmonic motion (SHM) based on that. Here’s a step-by-step solution:

### Step 1: Identify the Potential Energy Function
The potential energy function is given as:
\[ U(x) = a + b x^2 \]
where \( a \) and \( b \) are positive constants.

### Step 2: Calculate the Force
The force acting on the particle can be found using the relation:
\[ F = -\frac{dU}{dx} \]
Calculating the derivative:
\[ \frac{dU}{dx} = \frac{d}{dx}(a + b x^2) = 0 + 2bx = 2bx \]
Thus, the force becomes:
\[ F = -2bx \]

### Step 3: Relate Force to Mass and Acceleration
According to Newton's second law, the force can also be expressed as:
\[ F = m a \]
where \( a \) is the acceleration of the particle. Therefore, we have:
\[ ma = -2bx \]

### Step 4: Express Acceleration in Terms of Displacement
Acceleration \( a \) can be expressed in terms of displacement \( x \) as:
\[ a = \frac{d^2x}{dt^2} = -\omega^2 x \]
where \( \omega \) is the angular frequency of the motion. Substituting this into the force equation gives:
\[ m(-\omega^2 x) = -2bx \]

### Step 5: Simplify the Equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ -m\omega^2 = -2b \]
Thus, we find:
\[ m\omega^2 = 2b \]

### Step 6: Solve for Angular Frequency
From the equation above, we can express \( \omega^2 \):
\[ \omega^2 = \frac{2b}{m} \]
Taking the square root gives:
\[ \omega = \sqrt{\frac{2b}{m}} \]

### Step 7: Relate Angular Frequency to Frequency
The frequency \( f \) is related to the angular frequency \( \omega \) by the formula:
\[ f = \frac{\omega}{2\pi} \]
Substituting for \( \omega \):
\[ f = \frac{1}{2\pi} \sqrt{\frac{2b}{m}} \]

### Conclusion
From the final expression for frequency, we can see that the frequency \( f \) depends on both \( b \) and \( m \). Therefore, the correct answer is that the frequency is determined by the values of \( b \) and \( m \) alone.

### Final Answer
The frequency of the simple harmonic motion is determined by the values of \( b \) and \( m \) alone.

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