Question

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 $200\, N\, m^{-1}$. The maximum compression produced in the spring will be

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

Answer 1

The Correct Option is B

To solve this problem, we need to determine the maximum compression in the spring when a solid cylinder collides with it. We'll utilize the law of conservation of mechanical energy for this situation. 

The total mechanical energy before the collision will be the kinetic energy of the rolling cylinder, which includes both translational and rotational kinetic energy. After the collision, all this energy will be converted into the potential energy of the compressed spring.

Step-by-step Solution:

1. Calculate the translational kinetic energy \( KE_{\text{trans}} \) of the cylinder:
   • The formula for translational kinetic energy is: \(KE_{\text{trans}} = \frac{1}{2} m v^2\)
   • Given: \(m = 3\, \text{kg}\) and \(v = 4\, \text{m/s}\)
   • Calculate: \(KE_{\text{trans}} = \frac{1}{2} \times 3 \times (4)^2 = 24\, \text{J}\)
2. Calculate the rotational kinetic energy \( KE_{\text{rot}} \) of the cylinder:
   • For a solid cylinder, the moment of inertia \(I = \frac{1}{2} m R^2\). The angular velocity \( \omega \) is related to linear velocity by \(\omega = \frac{v}{R}\).
   • The formula for rotational kinetic energy is: \(KE_{\text{rot}} = \frac{1}{2} I \omega^2 = \frac{1}{2} \times \frac{1}{2} m R^2 \times \left(\frac{v}{R}\right)^2\)
   • Simplify to find: \(KE_{\text{rot}} = \frac{1}{4} m v^2\)
   • Calculate: \(KE_{\text{rot}} = \frac{1}{4} \times 3 \times (4)^2 = 12\, \text{J}\)
3. Total initial mechanical energy \( E_{\text{initial}} \):
   • \(E_{\text{initial}} = KE_{\text{trans}} + KE_{\text{rot}} = 24 + 12 = 36\, \text{J}\)
4. At maximum compression, all mechanical energy is converted into spring potential energy \( U_{\text{spring}} \):
   • The formula for spring potential energy is: \(U_{\text{spring}} = \frac{1}{2} k x^2\), where \( k = 200\, \text{N/m} \) is the spring constant and \( x \) is the compression.
5. Equate total energy to obtain: \(\frac{1}{2} k x^2 = 36\)
   • Solve for \( x \): \(100 x^2 = 36 \quad \Rightarrow \quad x^2 = \frac{36}{100} \quad \Rightarrow \quad x = \sqrt{\frac{36}{100}} = 0.6\, \text{m}\)

Therefore, the maximum compression produced in the spring is 0.6 m.