An ideal massless spring S can compressed `1.0` m in equilibrium by a force of `1000 N`. This same spring is placed at the bottom of a friction less inclined plane which makes an angle `theta =30^@` with the horizontal. A `10 kg` mass `m` is released from the rest at top of the inclined plane and is brought to rest momentarily after compressing the spring by `2.0 m.` the distance through which the mass moved before coming to rest is.

Options

A. `8 m`
B. `6 m`
C. `4 m`
D. `5 m`

Explanation

To solve the problem, we will use the principle of conservation of energy. The energy lost by the mass as it moves down the incline will be equal to the energy stored in the spring when it is compressed.

### Step-by-Step Solution:

1. **Determine the Spring Constant (k)**:
The spring constant \( k \) can be calculated using the information given. The spring is compressed by \( 1.0 \, \text{m} \) under a force of \( 1000 \, \text{N} \).
\[ k = \frac{F}{x} = \frac{1000 \, \text{N}}{1.0 \, \text{m}} = 1000 \, \text{N/m} \]

2. **Calculate the Potential Energy at the Top of the Incline**:
The mass \( m = 10 \, \text{kg} \) is released from rest. The height \( h \) from which it falls can be expressed in terms of the distance \( d \) it travels down the incline and the angle \( \theta = 30^\circ \):
\[ h = d \sin(30^\circ) = \frac{d}{2} \]
The potential energy at the top is given by:
\[ PE = mgh = mg \left(\frac{d}{2}\right) = 10 \cdot 9.81 \cdot \frac{d}{2} = 49.05d \, \text{J} \]

3. **Calculate the Energy Stored in the Spring**:
When the spring is compressed by \( 2.0 \, \text{m} \), the energy stored in the spring is:
\[ E_s = \frac{1}{2} k x^2 = \frac{1}{2} \cdot 1000 \cdot (2.0)^2 = \frac{1}{2} \cdot 1000 \cdot 4 = 2000 \, \text{J} \]

4. **Apply Conservation of Energy**:
The potential energy at the top of the incline will equal the energy stored in the spring when the mass comes to rest:
\[ PE = E_s \]
\[ 49.05d = 2000 \]

5. **Solve for d**:
\[ d = \frac{2000}{49.05} \approx 40.7 \, \text{m} \]

6. **Calculate Total Distance Travelled**:
The total distance travelled by the mass before coming to rest is:
\[ \text{Total Distance} = d + 2 = 40.7 + 2 = 42.7 \, \text{m} \]

### Final Answer:
The distance through which the mass moved before coming to rest is approximately \( 42.7 \, \text{m} \).