Question

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is 200 N/m. The block is pushed such that the length of the spring becomes 1 m and then released. At distance $ x $ m ($ x \leq 2 $) from the wall, the speed of the block will be:

• A. \( 10 \left[ 1 - (2 - x)^2 \right] \, \text{m/s} \)
• B. \( 10 \left[ 1 - (2 - x) \right]^{3/2} \, \text{m/s} \)
• C. \( 10 \left[ 1 - (2 - x)^2 \right]^{1/2} \, \text{m/s} \)
• D. \( 10 \left[ 1 - (2 - x)^2 \right]^2 \, \text{m/s} \)

Hint:

In spring-mass systems, use conservation of mechanical energy to relate potential energy and kinetic energy. The speed of the block can be found by equating the total energy at different points in the motion.

Answer 1

The Correct Option is C

To solve this problem, we need to use the conservation of mechanical energy principle as the force involved is conservative and there is no energy loss due to friction.

The mechanical energy in a spring-mass system is the sum of kinetic energy (due to motion) and potential energy (stored in the spring).

The initial configuration of the system is when the spring is compressed to 1 m, and the block is released from rest.

1. Initially, the spring is compressed from its natural length (2 m) to 1 m. So, the initial compression \(x_i = 2 - 1 = 1 \, \text{m}\).
2. The potential energy stored in the spring when compressed by \(x_i\) is given by: \(U_i = \frac{1}{2} k x_i^2 = \frac{1}{2} \times 200 \times 1^2 = 100 \, \text{J}\).
3. Initially, the block is at rest, so its kinetic energy \(K_i = 0\).
4. When the block is at a distance \(x\) from the wall, the spring is stretched by \(x_s = 2 - x\) (since 2 m is the natural length).
5. At this point, the potential energy of the spring is: \(U_f = \frac{1}{2} k x_s^2 = \frac{1}{2} \times 200 \times (2 - x)^2 \, \text{J}\).
6. Let the speed of the block at this point be \(v\). Then, its kinetic energy is: \(K_f = \frac{1}{2} m v^2\).
7. Using conservation of mechanical energy: \(U_i + K_i = U_f + K_f\).

Substituting the known values:

\(100 + 0 = \frac{1}{2} \times 200 \times (2 - x)^2 + \frac{1}{2} \times 2 \times v^2\)

Simplifying for \(v^2\):

\(100 = 100 (2 - x)^2 + v^2\) \(v^2 = 100 - 100 (2 - x)^2\) \(v^2 = 100 [1 - (2 - x)^2]\)

Thus, the speed \(v\) is:

\(v = 10 \sqrt{1 - (2 - x)^2}\)

Therefore, the speed of the block at a distance \(x\) from the wall is \(10 \left[ 1 - (2 - x)^2 \right]^{1/2} \, \text{m/s}\), which matches the correct option.

Answer 2

The total mechanical energy is the sum of the potential energy stored in the spring and the kinetic energy of the block. The potential energy \( U \) stored in a spring is given by: \( U = \frac{1}{2} k (x_{\text{spring}})^2 \), 
where \( k = 200 \, \text{N/m} \) is the spring constant, and \( x_{\text{spring}} \) is the displacement from the natural length of the spring. The initial displacement of the spring is 1 m (the block is compressed), so the initial potential energy is: 
\( U_{\text{initial}} = \frac{1}{2} \times 200 \times (2 - 1)^2 = 100 \, \text{J} \).

The total mechanical energy of the system is constant and is the sum of the kinetic energy \( K = \frac{1}{2} m v^2 \) and the potential energy stored in the spring at any point during the motion.<br>
The total energy \( E \) is given by: 
\( E = K + U \). 
At any position \( x \), the total energy is: 
\( E = \frac{1}{2} m v^2 + \frac{1}{2} k (2 - x)^2 \).

Since the total mechanical energy is conserved and the initial energy is all potential energy, we have: 
\( 100 = \frac{1}{2} \times 2 \times v^2 + \frac{1}{2} \times 200 \times (2 - x)^2 \). 
Simplifying: 
\( 100 = v^2 + 200 (2 - x)^2 \).

Solving for \( v \): 
\( v^2 = 100 - 200 (2 - x)^2 \), 
\( v = 10 \left[ 1 - (2 - x)^2 \right]^{1/2} \, \text{m/s} \).

Thus, the speed of the block at distance \( x \) from the wall is: 
\( \boxed{10 \left[ 1 - (2 - x)^2 \right]^{1/2}} \, \text{m/s} \).

Therefore, the correct answer is Option (3).