A 200-N sled slides down a frictionless hillside that rises at 37 degrees above the horizontal. What is the magnitude of the force that the hill exerts on the sled parallel to the surface of the hill?
A) 120 N
B) 0 N
C) 160 N
D) 150 N
E) 200 N
The correct answer and explanation is :
To solve this problem, we need to determine the magnitude of the force that the hill exerts on the sled parallel to the surface of the hill. We are given the weight of the sled, which is 200 N, and the angle of the hill, which is 37 degrees above the horizontal.
Step 1: Understanding the forces acting on the sled
There are two primary forces acting on the sled:
- Gravitational Force (Weight): The weight of the sled acts vertically downward and has a magnitude of 200 N.
- Normal Force: This is the force that the hill exerts on the sled perpendicular to its surface. It is responsible for preventing the sled from accelerating through the hill.
Since the hill is frictionless, we do not need to account for any frictional forces, and we will focus on the components of the gravitational force acting along and perpendicular to the surface of the hill.
Step 2: Resolving the weight into components
The gravitational force can be resolved into two components:
- Parallel to the surface of the hill: This component causes the sled to slide down the hill. It can be calculated using the formula:
[
F_{\parallel} = W \sin(\theta)
]
where:
- (W) is the weight of the sled (200 N),
- (\theta) is the angle of the hill (37°). So, the parallel component is:
[
F_{\parallel} = 200 \times \sin(37^\circ)
]
Using a calculator, (\sin(37^\circ) \approx 0.6018), so:
[
F_{\parallel} = 200 \times 0.6018 \approx 120.36 \, \text{N}
]
Step 3: Conclusion
The magnitude of the force that the hill exerts on the sled parallel to the surface is approximately 120 N. This matches option A.
Therefore, the correct answer is:
A) 120 N.
This force represents the component of gravity pulling the sled down the hill, and it is the force that the hill exerts on the sled along the surface.