what is the maximum speed with which a 1050kg car can round a turn of radius 70 m on a flat road if the coefficient of friction between tires and road is .80? is this result independent of the mass of the car?
m g (.8) = m v^2/r
m cancels
.8 g = v^2/r
To determine the maximum speed at which a car can round a turn, we need to consider the centripetal force acting on the car. The centripetal force is provided by the friction force between the tires and the road.
The maximum friction force can be calculated using the coefficient of friction (µ) and the normal force (N), which is the weight of the car (mg), where g is the acceleration due to gravity. The formula to calculate the maximum friction force (Fmax) is:
Fmax = µ * N
Now, let's calculate the maximum friction force. We know the coefficient of friction (µ = 0.80) and the weight of the car (m = 1050 kg, g = 9.8 m/s^2), so we can substitute these values into the formula:
Fmax = 0.80 * (1050 kg * 9.8 m/s^2)
Simplifying the equation:
Fmax = 0.80 * 10290 N
Fmax = 8232 N
Now, the centripetal force required to keep the car moving in a circle of radius 70 m can be calculated using the formula:
Fc = (mass * velocity^2) / radius
We want to find the maximum speed (velocity) at which the car can round the turn, so we set the centripetal force (Fc) equal to the maximum friction force (Fmax):
(mass * velocity^2) / radius = Fmax
Substituting the given values:
(1050 kg * velocity^2) / 70 m = 8232 N
Simplifying the equation:
velocity^2 = (70 m * 8232 N) / 1050 kg
velocity^2 = 549840 m^2/s^2 / 1050 kg
velocity^2 = 523.657 m^2/s^2
Taking the square root of both sides of the equation to find the velocity:
velocity ≈ √(523.657 m^2/s^2)
velocity ≈ 22.88 m/s
Therefore, the maximum speed at which the 1050 kg car can round the turn is approximately 22.88 m/s.
Now, to answer the second part of your question, whether this result is independent of the mass of the car: No, this result is not independent of the mass of the car. The maximum speed at which a car can round a turn depends on the mass of the car. As the mass increases, the centripetal force required to keep the car moving in a circle also increases. Thus, a heavier car would have a lower maximum speed compared to a lighter car on the same curve.