A race car travels 90 m/s around a circular
track of radius 93.5 m.
2.9 mg
µ = 0.12
19◦
What is the magnitude of the resultant
force on the 2900 kg driver and his car if
the car does not slip?
To find the magnitude of the resultant force on the driver and his car, we need to analyze the forces acting on the car as it travels around the circular track.
First, let's identify the forces involved:
1. Centripetal Force (Fc): This is the force directed towards the center of the circle, responsible for keeping the car moving in a circular path.
2. Frictional Force (Ff): This force acts tangentially to the circular path, opposing the forward motion of the car.
Since the car does not slip, the frictional force provides the centripetal force required to keep the car on the track. Therefore, we can equate these two forces.
The formula for centripetal force is given by:
Fc = (m * v^2) / r
Where:
m = mass of the car and the driver
v = velocity of the car
r = radius of the circular track
Converting the given values:
m = 2900 kg (mass of the driver and car)
v = 90 m/s (velocity)
r = 93.5 m (radius)
Plugging in the values into the formula, we get:
Fc = (2900 kg * (90 m/s)^2) / 93.5 m
Calculating the value of Fc:
Fc = (2900 kg * 8100 m^2/s^2) / 93.5 m
Fc = 251273.422 m * kg / s^2
Now, we equate the frictional force (Ff) with the centripetal force (Fc):
Ff = Fc
Since the coefficient of friction (µ) and the normal force (Fn) are not given in the question, we cannot calculate the exact value of Ff. However, we can calculate the maximum value of static friction using the formula:
Ff(max) = µ * Fn
Converting the given value:
µ = 0.12
Next, we need to calculate the normal force (Fn) acting on the car. The normal force is the force exerted by the track on the car, perpendicular to the track's surface. Here, the car's weight is counteracted by the normal force.
The formula for the normal force is:
Fn = m * g
Where:
m = mass of the car and the driver
g = acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the given mass value:
Fn = 2900 kg * 9.8 m/s^2
Calculating the value of Fn:
Fn = 28420 kg * m/s^2
Now, plugging in the maximum frictional coefficient (µ) and the calculated normal force (Fn) into the formula for maximum static friction (Ff(max)):
Ff(max) = µ * Fn
Ff(max) = 0.12 * 28420 kg * m/s^2
Finally, since Ff = Fc, the magnitude of the resultant force on the driver and his car is:
Magnitude of the resultant force = Ff(max) = Fc
Therefore, the magnitude of the resultant force cannot be determined without knowing the value of the normal force (Fn) or the specific coefficient of friction (µ) for the car and the track.