https://file.aiquickdraw.com/mathbot/file/1728349903217fc6blkl2.jpg

Let's break down and solve this problem step by step.

We are given:
- Mass of the block, \( m = 2.0 \, \text{kg} \)
- Length of the ramp, \( d = 1.36 \, \text{m} \)
- Angle of the ramp, \( \theta = 36.9^\circ \)
- Initial velocity at the top of the ramp, \( v_1 = 1.0 \, \text{m/s} \)
- Final velocity at the bottom of the ramp, \( v_2 = 3.0 \, \text{m/s} \)
- The unknown coefficient of friction, and we need to find the net work done by external forces.

### Step 1: Find the change in kinetic energy
The net work done on the block can be related to the change in kinetic energy, based on the work-energy principle:
\[
W_{\text{net}} = \Delta K = K_2 - K_1
\]
Where:
- \( K_1 = \frac{1}{2} m v_1^2 \) is the initial kinetic energy at the top.
- \( K_2 = \frac{1}{2} m v_2^2 \) is the final kinetic energy at the bottom.

Let's calculate these:
\[
K_1 = \frac{1}{2} (2.0 \, \text{kg}) (1.0 \, \text{m/s})^2 = 1.0 \, \text{J}
\]
\[
K_2 = \frac{1}{2} (2.0 \, \text{kg}) (3.0 \, \text{m/s})^2 = 9.0 \, \text{J}
\]

Thus, the change in kinetic energy is:
\[
\Delta K = K_2 - K_1 = 9.0 \, \text{J} - 1.0 \, \text{J} = 8.0 \, \text{J}
\]

### Step 2: Apply the work-energy principle
The work-energy theorem tells us that the net work done by external forces is equal to the change in kinetic energy:
\[
W_{\text{net}} = \Delta K = 8.0 \, \text{J}
\]

Thus, the net work done by the external forces is \( \boxed{8.0 \, \text{J}} \).

The correct answer is (b) \( + 8.0 \, \text{J} \), which matches the answer circled in the image.

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Would you like further details on any step, or do you have any questions about this solution?

Here are 5 related questions to explore:
1. How does the angle of the incline affect the net work done on the block?
2. How would the work done change if there was friction involved?
3. What would happen to the kinetic energy if the block started from rest?
4. How do you calculate the gravitational potential energy change in this scenario?
5. How does the work-energy principle apply in other scenarios like a car accelerating on a flat surface?

**Tip**: In problems like this, the work-energy theorem is a powerful tool to simplify calculations without needing to consider all the individual forces explicitly!

A 2.0 kg block slides down a 1.36 m ramp with unknown coefficient of friction. The ramp makes an angle of 36.9° with respect to the horizontal. The block has a speed of 1.0 m/s at the top and 3.0 m/s at the bottom. What is the net work done on the block by external forces as it slides down?

In this problem, a 2.0 kg block slides down a 1.36 m ramp at an angle of 36.9°, with speeds of 1.0 m/s at the top and 3.0 m/s at the bottom. Using the work-energy theorem, we calculate the net work done on the block. This solution involves concepts such as kinetic energy and energy conservation.

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