The kinetic energy of a particle executing S.H.M. is 16 J when it is at its mean position. If the mass of the particle is 0.32 kg , then what is the maximum velocity of the particle

Options

A. 5 m/s
B. 15 m/s
C. 10 m/s
D. 20 m/s

Answer

To find the maximum velocity of a particle executing Simple Harmonic Motion (S.H.M.), we can use the relationship between kinetic energy, mass, and velocity. Here’s a step-by-step solution:

### Step 1: Understand the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where:
- \( KE \) is the kinetic energy,
- \( m \) is the mass of the object, and
- \( v \) is the velocity of the object.

### Step 2: Identify the known values
From the problem, we know:
- The kinetic energy at the mean position, \( KE = 16 \, \text{J} \)
- The mass of the particle, \( m = 0.32 \, \text{kg} \)

### Step 3: Substitute the known values into the kinetic energy formula
We can rearrange the kinetic energy formula to solve for the maximum velocity \( v \):
\[ 16 = \frac{1}{2} \times 0.32 \times v^2 \]

### Step 4: Simplify the equation
First, multiply both sides by 2 to eliminate the fraction:
\[ 32 = 0.32 \times v^2 \]

### Step 5: Solve for \( v^2 \)
Now, divide both sides by 0.32:
\[ v^2 = \frac{32}{0.32} \]

### Step 6: Calculate \( v^2 \)
Calculating the right-hand side:
\[ v^2 = 100 \]

### Step 7: Find \( v \)
Now, take the square root of both sides to find \( v \):
\[ v = \sqrt{100} = 10 \, \text{m/s} \]

### Conclusion
The maximum velocity of the particle is:
\[ \boxed{10 \, \text{m/s}} \]