A sports car of mass 1400 kg (including the driver) crosses the rounded top of a hill (radius = 93 m) at 27 m/s.
1) Determine the normal force exerted by the road on the car.
2) Determine the normal force exerted by the car on the 75 kg driver.
3) Determine the car speed at which the normal force on the driver equals zero.
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To find the normal force exerted by the road on the car, we can use the concept of centripetal force. The normal force acts perpendicular to the surface of the road and provides the necessary centripetal force to keep the car moving in a circular path.
1) The centripetal force can be calculated using the formula:
F_c = (m*v^2) / r
where F_c is the centripetal force, m is the mass, v is the velocity, and r is the radius of the circular path.
Plugging in the values:
F_c = (1400 kg * (27 m/s)^2) / 93 m
F_c ≈ 1089.68 N
Since the normal force is equal to the centripetal force, the normal force exerted by the road on the car is approximately 1089.68 N.
To find the normal force exerted by the car on the driver, we need to consider that the normal force counteracts the weight of the driver.
2) Since the driver is inside the car, the gravitational force acting on the driver is:
F_g = m*d
where F_g is the gravitational force, m is the mass of the driver, and d is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values:
F_g = 75 kg * 9.8 m/s^2
F_g = 735 N
Therefore, the normal force exerted by the car on the driver is equal to the gravitational force, which is 735 N.
To find the car speed at which the normal force on the driver equals zero, we need to consider that the normal force will be zero when the gravitational force is equal to the centrifugal force.
3) Set the gravitational force equal to the centripetal force:
m*d = (m*v^2) / r
Rearranging the equation to solve for v:
v = sqrt(r * g)
where v is the speed, r is the radius, and g is the acceleration due to gravity.
Plugging in the values:
v = sqrt(93 m * 9.8 m/s^2)
v ≈ 30.44 m/s
Therefore, the car must be traveling at a speed of approximately 30.44 m/s for the normal force on the driver to be zero.