Calculate the speed of a car which can cross the oil spill of the curve without slipping sideways in the following case.
A high-speed test track for cars has a curved section — an arc of a circle of radius R= 425 m. The curved section is banked at angle θ = 22◦ from the horizontal to help the cars to stay in the road while moving at high speeds.The acceleration of gravity is 9.8 m/s^2.
One day, oil spills on the track making a few meters of the curved section frictionless. Calculate the speed v of a car which can cross the oil spill of the curve without slipping sideways.
The track on which the cars travel has a curved section with radius r=425 m. This is banked at 22 degrees to the horizontal to improve the stability of the cars and preventing them from slipping. When oil spills on the tracks there is no force of friction acting between the tires of the cars and the track. There are two forces acting on any car that travels on the track; one is the gravitational force of attraction due to the Earth and the other is a centrifugal force due the movement of the car in a circular path.
As the track is banked at 22 degrees, the gravitational force can be divided into two components, one perpendicular to the surface and another parallel to the surface acting downwards. The components of the centrifugal force are one one acting perpendicular to the surface and other parallel to the surface acting upwards. For the car to remain stable, the downward and upward components described earlier should be equal.
m*v^2*cos 22/r = m*g* sin 22
=>v^2*cos 22/r = g*sin 22
=> v^2 = r*g*tan 22
substituting the values given
v^2 = 425*9.8*tan 22
=> v^2 = 36.867
=> v = sqrt(36.867)
=> v = 6.07 m/s
A car can move at a maximum linear velocity of 6.07 m/s over the curved frictionless track without slipping.