A Mercedes-Benz 300SL (m = 1700 kg) is parked on a road that rises 15 degrees above the horizontal. What are the magnitudes of (a) the normal force and (b) the static frictional force that the ground exerts on the tires? Important: Assume that the road is higher up to the right and lower down to the left.
Wc = m*g = 1700kg * 9.8N/kg = 16,660 N.
= Wt. of car.
a. Fn=16,660*cos15 = 16.1 N.=Normal force.
b. Fp = 16660*sin15 = 4312 N. = Force
parallel to the road.
Fp-Fs = 0
4312-Fs = 0
Fs = 4312 N. = Force of static friction.
Well, the Mercedes-Benz 300SL must be feeling pretty "uphill" trying to figure out the normal and frictional forces. Let's help it out!
(a) The normal force is the force exerted by the ground perpendicular to the road surface. To find its magnitude, we need to resolve the gravitational force into components. The component of gravity perpendicular to the road is m * g * cos(θ), where θ is the angle of the road. Here, θ = 15 degrees.
So, the normal force = m * g * cos(15 degrees), where g is the acceleration due to gravity.
(b) The static frictional force is the force that opposes the motion and prevents the car from sliding down the hill. It acts parallel to the road. The maximum static frictional force can be found using the equation: fs ≤ μ * N, where μ is the coefficient of static friction and N is the normal force.
Since the car is parked, it means the static frictional force must equal the magnitude of the component of gravity parallel to the road, which is m * g * sin(θ).
So, the static frictional force = m * g * sin(15 degrees), where g is the acceleration due to gravity.
Now, let's calculate those forces to help the Mercedes-Benz 300SL feel more grounded!
To find the magnitudes of the normal force and the static frictional force exerted on the tires of the Mercedes-Benz 300SL, we need to consider the forces acting on the car in the vertical and horizontal directions.
(a) Magnitude of the normal force:
The normal force is the force exerted by a surface perpendicular to it. It acts in the vertical direction and is equal in magnitude but opposite in direction to the weight of the car. In this case, the weight of the car is given by the formula:
Weight = mass * acceleration due to gravity
Weight = m * g,
where m is the mass of the car and g is the acceleration due to gravity.
Given that the mass of the car is 1700 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 1700 kg * 9.8 m/s^2
Weight = 16660 N
Since the road is inclined at an angle of 15 degrees above the horizontal, the normal force will be less than the weight. The normal force can be found using trigonometry:
Normal force = Weight * cos(theta),
where theta is the angle of inclination.
Given that theta = 15 degrees, we can substitute the values into the equation:
Normal force = 16660 N * cos(15 degrees)
Normal force ≈ 15956 N
Therefore, the magnitude of the normal force exerted on the tires is approximately 15956 N.
(b) Magnitude of the static frictional force:
The static frictional force is the force that opposes the tendency of an object to slide along a surface. It acts parallel to the surface and prevents the car from sliding down the inclined road.
To find the magnitude of the static frictional force, we need to consider the maximum possible value of static friction, which can be given by the equation:
Maximum static frictional force = coefficient of static friction * normal force.
However, in this case, we are not given the coefficient of static friction between the car's tires and the road. Without that information, we cannot calculate the value of the static frictional force.
Therefore, we cannot determine the magnitude of the static frictional force without knowing the coefficient of static friction.
To find the magnitudes of the normal force and the static frictional force exerted on the Mercedes-Benz tires, we need to consider the forces acting on the car.
(a) The normal force is the force exerted by the ground perpendicular to the surface. It counteracts the weight of the car. We can find the normal force using the following formula:
Normal force = Weight × cos(θ)
where Weight is the force acting on the car due to gravity, and θ is the angle of the road relative to the horizontal.
Weight = mass × gravitational acceleration
= m × g
In this case, the mass of the car (m) is given as 1700 kg, and the gravitational acceleration (g) is approximately 9.8 m/s^2.
Thus, Weight = 1700 kg × 9.8 m/s^2
Next, we find the normal force:
Normal force = Weight × cos(θ)
= (1700 kg × 9.8 m/s^2) × cos(15°)
(b) The static frictional force opposes the tendency of the car to slide down the road. It acts parallel to the surface and prevents any relative motion between the tires and the road. The maximum static frictional force is given by:
Frictional force = coefficient of static friction × Normal force
To find the coefficient of static friction (μ), we need additional information. Assuming that the car is stationary, the maximum static frictional force is required to keep it from sliding down the road. In this case, the static frictional force would be equal to the weight of the car:
Frictional force = Weight
So, in order to determine the maximum static frictional force, we can use the weight of the car calculated earlier:
Frictional force = (1700 kg × 9.8 m/s^2) × cos(15°)
Using these equations, we can now calculate the magnitudes of the normal force and the static frictional force acting on the Mercedes-Benz tires.