A motorcycle is traveling up one side of a hill and down the other side. The crest of the hill is a circular arc with a radius of 35.5 m. Determine the maximum speed that the cycle can have while moving over the crest without losing contact with the road.
To determine the maximum speed that the motorcycle can have while moving over the crest without losing contact with the road, we need to consider the forces acting on the motorcycle at the top of the hill.
At the top of the hill, two forces are acting on the motorcycle: the gravitational force (mg) acting vertically downward and the normal force (N) acting perpendicular to the road surface. The normal force provides the necessary centripetal force to keep the motorcycle in circular motion.
The maximum speed can be determined by equating the centripetal force with the gravitational force. The centripetal force can be calculated as the product of the mass (m) of the motorcycle, the maximum speed (v), and the acceleration due to gravity (g), divided by the radius of the hill (r).
Centripetal force = mv^2 / r
Gravitational force = mg
Setting these two forces equal to each other, we have:
mv^2 / r = mg
Canceling out the mass (m) and rearranging the equation, we get:
v^2 = rg
Now we can substitute the values given in the problem:
r = 35.5 m
g ≈ 9.8 m/s^2
Plugging these values into the equation, we can solve for v. Taking the square root of both sides of the equation will give us the maximum speed:
v = √(rg)
v = √(35.5 * 9.8)
v ≈ √347.9
v ≈ 18.6 m/s
Therefore, the maximum speed that the motorcycle can have while moving over the crest without losing contact with the road is approximately 18.6 m/s.