A 180 g
 block is launched by compressing a spring of constant k=
200N/m
 a distance of 15 cm
. The spring is mounted horizontally, and the surface directly under it is frictionless. But beyond the equilibrium position of the spring end, the surface has coefficient of friction μ=0.27
. This frictional surface extends 85 cm
, followed by a frictionless curved rise, as shown in the figure.
After launch, where does the block finally come to rest? Measure from the left end of the frictional zone.

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.

A 180 g block is launched by compressing a spring of constant k = 200 N/m a distance of 15 cm. The spring is mounted horizontally, and the surface directly under it is frictionless. But beyond the equilibrium position of the spring end, the surface has coefficient of friction μ = 0.27. This frictional surface extends 85 cm, followed by a frictionless curved rise. After launch, where does the block finally come to rest? Measure from the left end of the frictional zone.

Explore how to solve a physics problem involving a spring-launched block with friction. Learn key concepts like energy conservation and the work-energy theorem. This problem involves calculating the distance at which a 180g block comes to rest after being launched by a spring with constant k = 200 N/m over a frictional surface. Suitable for Grade 11-12 physics students.

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