A car with a mass of 1400 kg is turning on a banked curve with a radius of 75 meters and an angle of 17°.
(a) Determine the ideal speed in which the car should take the turn. (Points: 5)
(b) If we assume 𝜇𝑠 between the tires and the road is 0.1, what is the MINIMUM speed the driver can take the turn? Draw a free-body diagram for full credit! (Points: 10)
To find the ideal speed for the car to take the turn, we can use the centripetal force equation. The centripetal force is equal to the friction force between the tires and the road.
(a) Determining the ideal speed:
1. Start by calculating the net force acting on the car as it turns.
The net force is the difference between the gravitational force acting downwards and the friction force acting upwards:
Net force = Gravitational force - Friction force
2. Calculate the gravitational force acting on the car.
The gravitational force is given by the formula:
Gravitational force = mass * acceleration due to gravity
Gravitational force = 1400 kg * 9.8 m/s^2
3. Calculate the friction force.
The friction force is given by the equation:
Friction force = coefficient of friction * normal force
The normal force is the perpendicular force exerted by the banked curve on the car. It is equal to the gravitational force acting downwards.
Normal force = Gravitational force
4. Substitute the values into the friction force equation:
Friction force = 0.1 * Gravitational force
5. Substitute the values into the net force equation:
Net force = Gravitational force - Friction force
6. Calculate the centripetal force required:
Centripetal force = mass * acceleration
The acceleration can be found using the formula:
Acceleration = velocity^2 / radius
7. Equate the centripetal force and net force:
Centripetal force = Net force
8. Solve the equation for velocity:
mass * acceleration = Gravitational force - Friction force
mass * velocity^2 / radius = Gravitational force - Friction force
velocity^2 = (radius * (Gravitational force - Friction force)) / mass
velocity = √((radius * (Gravitational force - Friction force)) / mass)
Plug in the values and calculate to find the ideal speed.
(b) Determining the minimum speed:
1. The minimum speed occurs when the static friction force is at its maximum, preventing the car from sliding off the curve. The static friction force can be calculated using the same formula:
Friction force = coefficient of friction * normal force
2. Instead of using the gravitational force as the normal force, we need to calculate the component of the gravitational force that is vertical to the incline.
Normal force = Gravitational force * cos(angle)
3. Calculate the friction force using the new normal force:
Friction force = 0.1 * (Gravitational force * cos(angle))
4. Calculate the net force using the new friction force:
Net force = Gravitational force - Friction force
5. Set the net force equal to the centripetal force:
Net force = Centripetal force
6. Solve for the minimum velocity using the same equation as before:
velocity^2 = (radius * (Gravitational force - Friction force)) / mass
Take the square root of the calculated value to determine the minimum speed.
Note: To draw a free-body diagram, you would include the forces acting on the car, such as the gravitational force, the normal force, and the friction force.