The problem 28 in the image involves a block on an inclined plane, a spring, and forces of friction. Let's break it down step by step.

Problem summary:

• Inclined plane at an angle of 30º.
• Block mass: 100 kg.
• Spring constant: 500 N/m.
• Initial compression: 0.3 m.
• Friction coefficient on the inclined plane: 0.27.
• We need to find how far up the plane the block will rise before it stops, after being released from rest.

Key Concepts:

1. Energy conservation: The spring's potential energy is initially stored and will convert to kinetic energy and work done against friction as the block moves up the plane.
2. Forces acting on the block:
   • Weight (gravitational force),
   • Frictional force,
   • Spring force.

Solution Outline:

1. Initial energy (spring potential energy): $U_s = \frac{1}{2} k x^2$ where $k = 500 \, \text{N/m}$ and $x = 0.3 \, \text{m}$.

2. Work done by friction: $W_f = \mu m g d \cos(\theta)$ where $\mu = 0.27$, $m = 100 \, \text{kg}$, $g = 9.8 \, \text{m/s}^2$, and $d$ is the distance traveled.

3. Work done by the gravitational force: $W_g = m g d \sin(\theta)$ where $\theta = 30^\circ$.

4. Energy conservation equation: $U_s = W_f + W_g$ We solve this equation for $d$, the distance the block moves up the plane.

Step-by-Step Calculations:

1. Spring potential energy: $U_s = \frac{1}{2} (500) (0.3)^2 = 22.5 \, \text{J}$

2. Friction work: $W_f = \mu m g d \cos(\theta) = 0.27 \times 100 \times 9.8 \times d \times \cos(30^\circ)$ $W_f \approx 229.5 d$

3. Gravitational work: $W_g = m g d \sin(\theta) = 100 \times 9.8 \times d \times \sin(30^\circ)$ $W_g = 490 d$

4. Energy equation: $22.5 = 229.5 d + 490 d$ Simplifying: $22.5 = 719.5 d$ Solving for $d$: $d = \frac{22.5}{719.5} \approx 0.0313 \, \text{m}$

So, the block will rise approximately 0.031 meters (3.13 cm) up the inclined plane before coming to rest.

Let me know if you'd like more details or clarifications!

Related Questions:

1. How would the answer change if the incline angle was 45º instead of 30º?
2. What would happen if the friction coefficient were doubled?
3. How would the distance change if the spring constant was 1000 N/m?
4. How much energy would be dissipated due to friction by the time the block comes to rest?
5. What is the maximum velocity of the block while it's moving?

Tip: Always check unit consistency when solving problems involving energy, work, and forces to avoid errors.