A 1000-kg Indy car travels around a curve banked at 25° to the horizontal. If the radius of the curve is 80 m, at what speed must the car be traveling if no friction is present?
To determine the required speed of the car, we need to balance the forces acting on it. In this case, the force of gravity and the centripetal force due to the banked curve.
1. Calculate the gravitational force acting on the car:
The formula for gravitational force is F_gravity = m * g, where m is the mass and g is the acceleration due to gravity.
F_gravity = 1000 kg * 9.8 m/s^2 = 9800 N
2. Calculate the centripetal force required to keep the car moving on the curve:
The centripetal force is given by the formula F_centripetal = (m * v^2) / r, where m is the mass, v is the velocity, and r is the radius of the curve.
Rearranging the equation and substituting known values:
F_centripetal = (m * v^2) / r
(m * v^2) = F_centripetal * r
v^2 = (F_centripetal * r) / m
v = sqrt[(F_centripetal * r) / m]
The centripetal force is directed towards the center of the curve and is equal to the normal force acting on the car.
The normal force is given by the formula F_normal = m * g * cos(theta), where theta is the angle of banking.
F_normal = 1000 kg * 9.8 m/s^2 * cos(25°) = 8924.23 N
Substituting the values into the previous equation:
v = sqrt[(F_normal * r) / m]
v = sqrt[(8924.23 N * 80 m) / 1000 kg]
v = sqrt[713938.4 Nm / 1000 kg]
v = sqrt[713.94 m^2/s^2]
3. Calculate the speed of the car:
v = sqrt(713.94) m/s
v ≈ 26.72 m/s
Therefore, the car must be traveling at approximately 26.72 m/s to navigate the banked curve without friction.