This problem involves energy conservation principles, as well as the work-energy theorem, to calculate where the block will come to rest. Let's break it down step by step.

Given:

• Mass of the block: $m = 180 \, \text{g} = 0.18 \, \text{kg}$
• Spring constant: $k = 200 \, \text{N/m}$
• Spring compression: $x = 0.15 \, \text{m}$
• Coefficient of kinetic friction: $\mu = 0.27$
• Length of frictional zone: $L = 0.85 \, \text{m}$

Step 1: Calculate the potential energy stored in the compressed spring.

The potential energy $E_{\text{spring}}$ stored in the spring when compressed is given by the formula: $E_{\text{spring}} = \frac{1}{2} k x^2$ Substitute the given values: $E_{\text{spring}} = \frac{1}{2} \times 200 \, \text{N/m} \times (0.15 \, \text{m})^2$

Step 2: Calculate the work done by friction.

The work done by friction is given by: $W_{\text{friction}} = F_{\text{friction}} \times d = \mu m g d$ where:

• $\mu = 0.27$ is the coefficient of friction,
• $m = 0.18 \, \text{kg}$ is the mass,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $d$ is the distance traveled on the frictional surface (this is the value we need to find).

We assume the block stops when the work done by friction equals the initial spring energy. Thus: $\frac{1}{2} k x^2 = \mu m g d$ Solve for $d$.

I'll calculate the final value.The block will come to rest approximately 4.72 meters from the left end of the frictional zone.

Summary of steps:

• The spring's potential energy is converted into kinetic energy.
• As the block moves across the frictional surface, friction reduces its kinetic energy until it comes to rest after traveling 4.72 meters.

Would you like further details, or do you have any questions?

Here are 5 related questions to expand on this concept:

1. How would the result change if the spring constant were doubled?
2. What happens if the block's mass is increased while keeping all other factors the same?
3. Can we calculate the speed of the block when it enters the frictional zone?
4. How much energy is lost due to friction?
5. What is the role of the frictionless curved rise in this scenario?

Tip: Always convert units to standard SI units (meters, kilograms, etc.) when working with physics equations to avoid calculation errors.