A solid cylinder of mass `3 kg` is rolling on a horizontal surface with velocity `4 ms^(-1)`. It collides with a horizontal spring of force constant `200 Nm^(-1)`. The maximum compression produced in the spring will be `:`

Options

A. 0.7 m
B. 0.2 m
C. 0.5 m
D. 0.6 m

Explanation

To solve the problem of finding the maximum compression produced in the spring when a solid cylinder collides with it, we can follow these steps:

### Step 1: Identify the given values
- Mass of the cylinder, \( m = 3 \, \text{kg} \)
- Velocity of the cylinder, \( v = 4 \, \text{m/s} \)
- Spring constant, \( k = 200 \, \text{N/m} \)

### Step 2: Calculate the initial kinetic energy of the cylinder
The total kinetic energy \( KE \) of a rolling solid cylinder is given by the sum of its translational and rotational kinetic energy:

\[ KE = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 \]

For a solid cylinder, the moment of inertia \( I \) is given by:

\[ I = \frac{1}{2} m r^2 \]

And the relationship between linear velocity \( v \) and angular velocity \( \omega \) is:

\[ \omega = \frac{v}{r} \]

Substituting \( \omega \) into the kinetic energy equation:

\[ KE = \frac{1}{2} m v^2 + \frac{1}{2} \left(\frac{1}{2} m r^2\right) \left(\frac{v}{r}\right)^2 \]

This simplifies to:

\[ KE = \frac{1}{2} m v^2 + \frac{1}{4} m v^2 = \frac{3}{4} m v^2 \]

### Step 3: Substitute the values into the kinetic energy formula
Now, substituting the values of \( m \) and \( v \):

\[ KE = \frac{3}{4} \times 3 \, \text{kg} \times (4 \, \text{m/s})^2 \]

Calculating this gives:

\[ KE = \frac{3}{4} \times 3 \times 16 = \frac{144}{4} = 36 \, \text{J} \]

### Step 4: Set up the energy conservation equation
At maximum compression \( x \), the kinetic energy will be converted into the potential energy stored in the spring:

\[ \frac{1}{2} k x^2 = KE \]

Substituting \( KE \) into the equation:

\[ \frac{1}{2} k x^2 = 36 \]

### Step 5: Solve for \( x^2 \)
Rearranging the equation to solve for \( x^2 \):

\[ x^2 = \frac{2 \times 36}{k} = \frac{72}{200} = 0.36 \]

### Step 6: Calculate \( x \)
Taking the square root to find \( x \):

\[ x = \sqrt{0.36} = 0.6 \, \text{m} \]

### Conclusion
The maximum compression produced in the spring is:

\[ \boxed{0.6 \, \text{m}} \]