a 1710 kg passes over a hill that follows the arc of a circle of radius 41.5 meters. what is the maximum speed the car can have passing this point so the car does not leave the road
mg=mv^2/r
solve for v.
mass does not matter
m v^2/R = m g
so
v^2/R = g
v^2 = 9.81* 41.5
v = 20.2 m/s
To find the maximum speed at which the car can pass the hill without leaving the road, we can use the concept of centripetal force.
The centripetal force required to keep the car moving in a circular path is provided by the friction between the car's tires and the road. When the car is at the highest point on the hill, the gravitational force pulling it down is at its maximum. At this point, the gravitational force must be balanced by the centripetal force to prevent the car from leaving the road.
We can start by calculating the gravitational force acting on the car at the top of the hill. The weight force is given by:
Weight = mass * gravity
where mass = 1710 kg (given) and gravity = 9.8 m/s^2 (approximate value).
Weight = 1710 kg * 9.8 m/s^2
Weight = 16,758 N
Now, we can calculate the maximum allowable centripetal force. At the top of the hill, the frictional force between the tires and the road should provide this centripetal force. The frictional force is given by:
Frictional force = mass * centripetal acceleration
The centripetal acceleration is obtained from the circular motion equation:
centripetal acceleration = (velocity^2) / radius
Since we want to find the maximum speed, we assume the car barely stays on the road and the frictional force between the tires and the road is at its maximum. At this point, the maximum frictional force can be found using the equation:
maximum frictional force = coefficient of static friction * normal force
Now, the normal force can be calculated as:
Normal force = Weight - force exerted by the car down the hill
At the top of the hill, the force exerted by the car down the hill is equal to the weight force acting on it. Therefore:
Normal force = Weight - Weight
Normal force = 0 N
Substituting the normal force into the maximum frictional force equation, we get:
maximum frictional force = coefficient of static friction * 0 N
maximum frictional force = 0 N
This means that no frictional force is available to provide the required centripetal force. Hence, the car cannot pass the hill without leaving the road, regardless of its speed.