To determine the maximum compression of the spring, we can use the principle of conservation of energy. Initially, the solid cylinder has kinetic energy due to its translational and rotational motion. As it compresses the spring, this kinetic energy is converted into elastic potential energy stored in the spring.
Here's how you can solve it:
1. **Calculate the initial kinetic energy of the cylinder:**
The total kinetic energy (\( KE \)) of a rolling cylinder is the sum of its translational kinetic energy (\( KE_{trans} \)) and rotational kinetic energy (\( KE_{rot} \)).
The translational kinetic energy is given by:
\[
KE_{trans} = \frac{1}{2} m v^2
\]
where \( m \) is the mass of the cylinder (3 kg) and \( v \) is its velocity (4 m/s).
\[
KE_{trans} = \frac{1}{2} \times 3 \, \text{kg} \times (4 \, \text{m/s})^2 = \frac{1}{2} \times 3 \times 16 = 24 \, \text{J}
\]
The rotational kinetic energy for a solid cylinder is given by:
\[
KE_{rot} = \frac{1}{2} I \omega^2
\]
where \( I \) is the moment of inertia of the cylinder and \( \omega \) is its angular velocity.
For a solid cylinder, \( I = \frac{1}{2} m r^2 \) and \( \omega = \frac{v}{r} \). Thus:
\[
KE_{rot} = \frac{1}{2} \times \frac{1}{2} m r^2 \times \left(\frac{v}{r}\right)^2 = \frac{1}{4} m v^2
\]
\[
KE_{rot} = \frac{1}{4} \times 3 \, \text{kg} \times (4 \, \text{m/s})^2 = \frac{1}{4} \times 3 \times 16 = 12 \, \text{J}
\]
So, the total initial kinetic energy is:
\[
KE_{total} = KE_{trans} + KE_{rot} = 24 \, \text{J} + 12 \, \text{J} = 36 \, \text{J}
\]
2. **Equate the total initial kinetic energy to the elastic potential energy stored in the spring at maximum compression:**
The elastic potential energy (\( E_{spring} \)) of the spring is given by:
\[
E_{spring} = \frac{1}{2} k x^2
\]
where \( k \) is the spring constant (200 N/m) and \( x \) is the maximum compression of the spring.
Set this equal to the total kinetic energy:
\[
\frac{1}{2} k x^2 = KE_{total}
\]
\[
\frac{1}{2} \times 200 \, \text{N/m} \times x^2 = 36 \, \text{J}
\]
\[
100 \, x^2 = 36
\]
\[
x^2 = \frac{36}{100} = 0.36
\]
\[
x = \sqrt{0.36} = 0.6 \, \text{m}
\]
So, the maximum compression of the spring is **0.6 m**. Therefore, the correct answer is **D. 0.6 m**.
