A 184 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.

To solve this problem, we can break it down into several steps based on the conservation of energy and the work done by friction.

### Step 1: Energy stored in the spring
The block is launched by compressing the spring. The potential energy stored in the spring when it is compressed is:

\[
E_{\text{spring}} = \frac{1}{2} k x^2
\]

where:
- \( k = 200 \, \text{N/m} \) (spring constant),
- \( x = 0.15 \, \text{m} \) (compression distance of the spring).

Substitute the values:

\[
E_{\text{spring}} = \frac{1}{2} (200) (0.15)^2 = 2.25 \, \text{J}
\]

This is the energy available to the block as it is launched.

### Step 2: Work done by friction
After the block is launched, it travels through a frictionless surface until it reaches the frictional zone. When the block enters the region with friction, the kinetic energy is gradually lost due to the work done by friction.

The work done by friction is given by:

\[
W_{\text{friction}} = f \cdot d
\]

where:
- \( f = \mu mg \) is the frictional force,
- \( d = 0.85 \, \text{m} \) is the distance over which friction acts,
- \( \mu = 0.27 \) is the coefficient of friction,
- \( m = 0.184 \, \text{kg} \) is the mass of the block,
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity.

The frictional force is:

\[
f = \mu mg = 0.27 \times 0.184 \times 9.8 = 0.487 \, \text{N}
\]

Thus, the work done by friction over the 0.85 m frictional zone is:

\[
W_{\text{friction}} = 0.487 \times 0.85 = 0.414 \, \text{J}
\]

### Step 3: Net energy available after friction
The initial energy from the spring is 2.25 J, and the friction removes 0.414 J from the block's energy. So, the remaining energy is:

\[
E_{\text{remaining}} = 2.25 - 0.414 = 1.836 \, \text{J}
\]

This remaining energy is the kinetic energy the block has after leaving the frictional zone.

### Step 4: Motion on the frictionless rise
Now, the block moves onto the frictionless curved rise. The kinetic energy is converted to potential energy. The height \( h \) where the block finally stops is determined by setting the remaining kinetic energy equal to the gravitational potential energy at height \( h \):

\[
E_{\text{remaining}} = mgh
\]

Substitute the values:

\[
1.836 = 0.184 \times 9.8 \times h
\]

Solving for \( h \):

\[
h = \frac{1.836}{0.184 \times 9.8} = 1.02 \, \text{m}
\]

Thus, the block will rise to a height of **1.02 meters** on the frictionless curve before coming to rest.

### Conclusion:
The block will come to rest at a height of 1.02 meters after traveling through the frictional zone and the frictionless curved rise.

Do you have any questions or would you like further details?

Here are 5 questions related to this problem:
1. How would the result change if the coefficient of friction were doubled?
2. What would happen if the spring compression was halved?
3. How does the spring constant affect the block's final stopping point?
4. What would be the effect if the surface under the spring was also frictional?
5. How can we calculate the speed of the block when it enters the frictional zone?

**Tip**: Always break energy problems into conservation and work sections to better visualize how energy is transferred in stages.

A 184 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 the solution to a physics problem involving a 184 g block launched by a spring (k = 200 N/m) compressed by 15 cm. The block moves through a frictional zone with μ = 0.27 and a length of 85 cm before climbing a frictionless rise. Learn how energy conservation and friction affect the block's final resting point.

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