A car of mass 1396 kg is parked on a very slippery hillside (see figure below). To keep it from sliding down the hill (è = 15°), the owner attaches a cable. If there is no frictional force between the road and the tires, what is the tension in the cable?
T=mgsinα
To find the tension in the cable, we need to analyze the forces acting on the car.
First, let's draw a free body diagram:
|-- Mg --|
| |
-----------------------
| |
| |
| |
| Car |
| |
| |
|-----------------|
| |
|-------- T -------|
The forces acting on the car are:
1. The weight of the car (Mg), acting vertically downwards.
2. The tension in the cable (T), acting horizontally.
3. The normal force (N) from the hillside, acting perpendicular to the hillside.
Since there is no friction between the road and the tires, the normal force (N) will be equal to the weight of the car (Mg). However, we need to determine the component of the weight that is acting parallel to the hillside.
The component of the weight parallel to the hillside is given by: W_parallel = Mg * sin(θ), where θ is the angle of inclination (15° in this case).
Now we can write the equation for the force balance in the horizontal direction:
T - W_parallel = 0
Substituting the value of W_parallel, we get:
T - Mg * sin(θ) = 0
Rearranging the equation to solve for T, we have:
T = Mg * sin(θ)
Using the given values:
Mass of the car (M) = 1396 kg
Angle of inclination (θ) = 15°
Acceleration due to gravity (g) = 9.8 m/s^2
Let's calculate the tension in the cable:
T = Mg * sin(θ)
= 1396 kg * 9.8 m/s^2 * sin(15°)
≈ 2051.13 N
Therefore, the tension in the cable is approximately 2051.13 N.
To find the tension in the cable, we need to analyze the forces acting on the car.
Let's start by drawing a free body diagram for the car on the hillside:
```
|\
| \
| \
| \
F | \__
| / |
| / |
| / |
| / |
|/ |
```
In this diagram, F represents the force exerted by the cable, and the arrow represents the weight of the car, which acts vertically downward. To keep the car from sliding down the hill, the force exerted by the cable needs to balance the component of the weight that is acting down the hill.
The weight of the car can be calculated using the formula:
Weight = mass * gravitational acceleration
where the mass is given as 1396 kg and the gravitational acceleration is approximately 9.8 m/s².
Weight = 1396 kg * 9.8 m/s² = 13640.8 N
Now, we can calculate the component of the weight acting down the hill. This can be found using trigonometry since we know the angle of inclination (15°). The component of the weight down the hill is given by:
Component down the hill = Weight * sin(angle)
Component down the hill = 13640.8 N * sin(15°) ≈ 3508.6 N
Since the cable exerts an equal and opposite force to balance this component down the hill, the tension in the cable is also 3508.6 N.
Therefore, the tension in the cable is approximately 3508.6 N.