A motorcycle has a constant speed of 21.6 m/s as it passes over the top of a hill whose radius of curvature is 100 m. The mass of the motorcycle and driver is 392 kg. Find the magnitude of (a) the centripetal force and (b) the normal force that acts on the cycle.
To find the magnitude of the centripetal force and the normal force acting on the motorcycle, we need to use the equations for centripetal force and gravitational force.
(a) Centripetal force:
The centripetal force is the force that keeps an object moving in a curved path. It is directed towards the center of the circular motion. The formula to calculate centripetal force is:
Fc = (m × v²) / r
Where:
Fc = centripetal force
m = mass of the object (392 kg)
v = velocity of the object (21.6 m/s)
r = radius of curvature of the hill (100 m)
Substituting the values into the formula, we get:
Fc = (392 kg × (21.6 m/s)²) / 100 m
Calculating this, we find:
Fc ≈ 3639.85 N
So, the magnitude of the centripetal force acting on the motorcycle is approximately 3639.85 N.
(b) Normal force:
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force opposes the force due to gravity. The magnitude of the normal force can be calculated using the formula:
F_normal = mg + Fc
Where:
F_normal = normal force
m = mass of the object (392 kg)
g = acceleration due to gravity (approximately 9.8 m/s²)
Fc = centripetal force (which we found to be approximately 3639.85 N)
Substituting the values into the formula, we get:
F_normal = (392 kg × 9.8 m/s²) + 3639.85 N
Calculating this, we find:
F_normal ≈ 7815.65 N
Therefore, the magnitude of the normal force acting on the motorcycle is approximately 7815.65 N.