a. Find the magnitude of the force required to keep a 3100-pound car from sliding down a hill inclined at 5.7° from the horizontal.
b. Find the magnitude of the force of the car against the hill.
component of weight down slope = w * sin 5.7
component of weight perpendicular to slope = w * cos 5.7
a. To find the magnitude of the force required to keep the car from sliding down the hill, we can use the concept of gravitational force and the angle of inclination.
The component of the gravitational force acting parallel to the hill will try to make the car slide down the hill. This component can be calculated using the formula:
Force parallel to the hill = weight of the car * sin(angle of inclination)
Given that the weight of the car is 3100 pounds, and the angle of inclination is 5.7 degrees, we can calculate the force required as:
Force parallel to the hill = 3100 * sin(5.7°)
Using a calculator, we find:
Force parallel to the hill = 3100 * 0.0998 ≈ 309.38 pounds
Therefore, the magnitude of the force required to keep the car from sliding down the hill is approximately 309.38 pounds.
b. To find the magnitude of the force of the car against the hill, we need to find the component of the gravitational force perpendicular to the hill.
The component of the gravitational force acting perpendicular to the hill will push the car against the hill. This component can be calculated using the formula:
Force perpendicular to the hill = weight of the car * cos(angle of inclination)
Given that the weight of the car is 3100 pounds, and the angle of inclination is 5.7 degrees, we can calculate the force against the hill as:
Force perpendicular to the hill = 3100 * cos(5.7°)
Using a calculator, we find:
Force perpendicular to the hill = 3100 * 0.9949 ≈ 3081.59 pounds
Therefore, the magnitude of the force of the car against the hill is approximately 3081.59 pounds.
To find the magnitude of the force required to keep the car from sliding down the hill (part a), you need to use the concept of static friction. The maximum static friction force can be determined using the formula:
F_friction = μ_s *N
where F_friction is the friction force, μ_s is the coefficient of static friction, and N is the normal force.
1. Begin by finding the normal force (N) acting on the car. In this case, the normal force is equal to the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity. Convert the weight of the car from pounds to its mass in pounds (1 pound = 0.4536 kg).
mass = 3100 pounds * 0.4536 kg/pound
2. Next, calculate the normal force (N) using the formula:
N = mass * g
where g is approximately 9.8 m/s².
3. Now, you need to find the coefficient of static friction (μ_s) between the car's tires and the hill. Unfortunately, this information is usually unavailable. However, a typical value for the coefficient of static friction between rubber tires and dry asphalt is around 0.7. You can adjust the value based on any other information provided.
4. Finally, compute the friction force (F_friction) using the formula:
F_friction = μ_s * N
This will give you the magnitude of the force required to keep the car from sliding down the hill.
To find the magnitude of the force of the car against the hill (part b), you can use the concept of the normal force.
5. Recall that the normal force (N) is already calculated in part a.
The magnitude of the force of the car against the hill is equal to the normal force (N).
Feel free to plug in the given values and perform the calculations to obtain the final answers.