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 car on the 74 kg driver. Determine the car speed at which the normal force on the driver equals zero.
Let the normal force exerted by the seat on the driver be F.
M g - F = M a = M V^2/R
F = M (g - V^2/R)
Use that equation to compute F when V = 23 m/s.
The mass cancels out.
For the last part, solve for V when F = 0.
V = sqrt(Rg)
To solve this problem, we need to consider the forces acting on the sports car and the driver at the top of the hill.
1. Determine the normal force exerted by the car on the driver:
At the top of the hill, the car and driver are in circular motion, experiencing a centripetal force. The net force acting towards the center of the circle is given by:
F_net = m * (v^2 / r)
Where:
F_net is the net force,
m is the mass of the car and driver (1300 kg),
v is the velocity of the car (23 m/s),
and r is the radius of the circular path (86 m).
Since the normal force (N) is the force exerted by the car on the driver in the vertical direction, it can be calculated as the difference between the weight (mg) and the centripetal force (F_net). The weight is given by:
Weight = m * g
Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, the normal force is:
N = Weight - F_net
1. Substitute the known values into the equations:
Weight = (Mass of car + Mass of driver) * g
= (1300 kg + 74 kg) * 9.8 m/s^2
Next, calculate the net force:
F_net = m * (v^2 / r)
2. Calculate the normal force:
N = Weight - F_net
2. Determine the car speed at which the normal force on the driver equals zero:
The normal force on the driver will equal zero when the net inward force (F_net) becomes equal to the weight (mg).
1. Set F_net equal to Weight:
F_net = Weight
2. Substitute the known values into the equation:
m * (v^2 / r) = m * g
3. Cancel out the mass (m) from both sides of the equation:
(v^2 / r) = g
4. Solve for the velocity (v):
v^2 = r * g
v = sqrt(r * g)
3. Substitute the values of r (86 m) and g (9.8 m/s^2) into the equation above to find the car speed.