Let's go through this problem step-by-step, breaking it down into parts as specified.

Given Data and Setup

1. Mass of the block, $m = 2 \, \text{kg}$.
2. Inclined plane length, $L = 4 \, \text{m}$.
3. Angle of inclination, $\theta = 30^\circ$.
4. Spring constant, $k = 100 \, \text{N/m}$.
5. Coefficient of friction for part (b), $\mu = 0.2$ (only relevant for part b).

The problem is broken into three parts:

1. Part (a): Maximum Compression of the Spring (Neglecting Friction)

   The block slides down the inclined plane, converting its gravitational potential energy at the top of the incline into kinetic energy and, ultimately, into elastic potential energy when it compresses the spring.

   • Gravitational Potential Energy (GPE) at the top: $\text{GPE} = m \cdot g \cdot h$ where $h = L \sin \theta$.

   • Elastic Potential Energy (EPE) at maximum compression: $\text{EPE} = \frac{1}{2} k x^2$ where $x$ is the compression of the spring.

   Equating GPE and EPE gives: $m \cdot g \cdot (L \sin \theta) = \frac{1}{2} k x^2$

2. Part (b): Maximum Compression of the Spring (Including Friction)

   Now, with friction, we need to account for the work done against friction as the block slides down the incline. This work is: $W_{\text{friction}} = \mu \cdot m \cdot g \cdot \cos \theta \cdot L$

   The modified energy equation is: $m \cdot g \cdot (L \sin \theta) - \mu \cdot m \cdot g \cdot \cos \theta \cdot L = \frac{1}{2} k x^2$

3. Part (c): Distance the Block Moves Up the Incline After Leaving the Spring

   After compressing the spring maximally, the block will be pushed back up the incline. The initial energy for this part is the elastic potential energy in the spring, which will be converted into gravitational potential energy as the block moves back up.

   Using conservation of energy: $\frac{1}{2} k x^2 = m \cdot g \cdot h_{\text{new}}$ where $h_{\text{new}} = d \sin \theta$ and $d$ is the distance up the incline we need to find.

Solution Steps

I'll calculate each part systematically.

Would you like me to proceed with the calculations, or do you have questions about any of these steps?

Relative Questions

1. How does friction affect the maximum compression of the spring?
2. Why is gravitational potential energy converted into elastic potential energy in this problem?
3. What role does the angle of the incline play in the calculations?
4. How would the results change if the spring constant were different?
5. Can we use energy conservation if friction is present?

Tip

Always remember to check if energy conservation applies directly in problems with friction; sometimes you may need to account for work done by friction separately.