A Mercedes-Benz 300SL (m = 1700 kg) is parked on a road that rises 15° 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?
To find the magnitudes of the normal force and the static frictional force on the car tires, we need to consider the forces acting on the car.
(a) The normal force (N) is the force exerted by a surface perpendicular to the object. In this case, it is the force exerted by the road on the car, pointing vertically upwards.
(b) The static frictional force (f) is the force that opposes the motion of an object in contact with a surface and is parallel to the surface. In this case, it is the force exerted by the road on the car, pointing parallel to the road surface.
To solve for these forces, we need to consider the forces acting along the vertical and horizontal directions.
1. Vertical Forces:
- Weight (mg): The weight of the car acts vertically downwards and is given by the formula: weight = mass × acceleration due to gravity.
weight = m × g = 1700 kg × 9.8 m/s².
Since the road is inclined at an angle of 15° above the horizontal, we need to resolve this weight into two components:
- The normal force (N) acts perpendicular to the inclined surface.
N = weight × cos(15°).
- The parallel force (P) acts parallel to the inclined surface.
P = weight × sin(15°).
2. Horizontal Forces:
- The static frictional force (f) opposes the motion of the car and acts parallel to the road, preventing the car from sliding down or up the incline. The maximum static frictional force is given by f = μs × N, where μs is the coefficient of static friction.
In this case, because the car is parked, it is not moving, so the static friction force prevents it from sliding down. Therefore, the maximum static frictional force is equal to the parallel force (P) acting on it.
The normal force and the static frictional force are then:
(a) Normal force (N) = weight × cos(15°).
(b) Static frictional force (f) = P = weight × sin(15°).
To complete the calculation, plug in the given values and evaluate the expressions.