A sports car of mass 1300 kg (including the driver) crosses the rounded top of a hill (r = 86 m) at 23 m/s. Determine the normal force exerted by the road on the car. Determine the normal force exerted by the car on the 74 kg driver. Determine the car speed at which the normal force on the driver equals zero.
Ac = v^2/r = 23^2/86 = 6.15 m/s^2
Force of gravity down on car = 1300*9.8 = 12740 N
Force of gravity - force up from road = m a = 1300*6.15 = 7995 N
so
force up from road = 12740 - 7995 = 4745 N
That should get you started. The rest of the problem is the same idea.
To determine the normal force exerted by the road on the car, we can use the concept of centripetal force. At the top of the hill, the normal force and gravitational force provide the centripetal force required for circular motion.
1. Calculate the gravitational force acting on the car:
Since we know the mass of the car is 1300 kg, we can determine the force of gravity acting on it. The acceleration due to gravity is approximately 9.8 m/s².
Gravitational force = mass * acceleration due to gravity
Gravitational force = 1300 kg * 9.8 m/s² = 12740 N
2. Calculate the centripetal force:
At the top of the hill, the centripetal force can be determined using the formula:
Centripetal force = (mass * velocity²) / radius
Centripetal force = (1300 kg * (23 m/s)²) / 86 m = 8769.77 N
3. Determine the normal force exerted by the road:
At the top of the hill, the normal force exerted by the road on the car is the difference between the gravitational force and the centripetal force:
Normal force = Gravitational force - Centripetal force
Normal force = 12740 N - 8769.77 N = 3970.23 N
To determine the normal force exerted by the car on the driver, we need to consider the relation between the driver's weight and the normal force on the driver.
4. Calculate the weight of the driver:
Weight = mass * acceleration due to gravity
Weight = 74 kg * 9.8 m/s² = 725.2 N
5. Determine the normal force exerted by the car on the driver:
At the top of the hill, the normal force exerted by the car on the driver is equal to the weight of the driver:
Normal force = Weight = 725.2 N
Finally, to determine the car's speed at which the normal force on the driver equals zero, we need to consider the maximum height of the hill.
6. Calculate the gravitational potential energy at the top of the hill:
At the top of the hill, the gravitational potential energy is equivalent to the kinetic energy of the car:
Gravitational potential energy = (mass * velocity²) / 2
Gravitational potential energy = (1300 kg * (23 m/s)²) / 2 = 706150 J
The gravitational potential energy can be calculated using the formula:
Gravitational potential energy = mass * gravitational acceleration * height
706150 J = 1300 kg * 9.8 m/s² * height
height = 706150 J / (1300 kg * 9.8 m/s²)
height ≈ 57.45 m
At the bottom of the hill, when the height is zero, the normal force on the driver would be zero. Therefore, the car speed at which the normal force on the driver equals zero is achieved when the car reaches the bottom of the hill.