A racetrack has the shape of an inverted cone, as the drawing shows. On this surface the cars race in circles that are parallel to the ground, and the surface is at an angle è = 20°. For a speed of 40 m/s, at what value of the distance d should a driver locate his car if he wishes to stay on a circular path without depending on friction?

Question

A racetrack has the shape of an inverted cone, as the drawing shows. On this surface the cars race in circles that are parallel to the ground, and the surface is at an angle è = 20°. For a speed of 40 m/s, at what value of the distance d should a driver locate his car if he wishes to stay on a circular path without depending on friction?

Answer 1

To answer this question, we need to understand the forces acting on the car and use the concept of centripetal acceleration.

In this case, the car is moving in a circular path without depending on friction. This means that the only force acting on the car is the gravitational force, which provides the centripetal acceleration necessary to keep the car on the circular path.

First, let's find the centripetal acceleration using the formula:

a = v^2 / r

where:
a is the centripetal acceleration,
v is the speed of the car, and
r is the radius of the circular path.

Given that the speed of the car is 40 m/s, we can calculate the centripetal acceleration by substituting the values into the formula:

a = (40 m/s)^2 / r

Now, the gravitational force provides the centripetal force. The gravitational force is given by:

F_gravity = m * g

where:
F_gravity is the gravitational force,
m is the mass of the car, and
g is the acceleration due to gravity (approximately 9.8 m/s^2).

Since the gravitational force is acting perpendicular to the surface of the racetrack, we can consider only the component of the gravitational force that is parallel to the circular path. This component is given by:

F_parallel = F_gravity * sin(è)

where:
è is the angle of the surface (20° in this case).

Now, since F_parallel is the centripetal force, we can equate it to the centripetal acceleration multiplied by the mass of the car:

F_parallel = m * a

Substituting the expressions for F_parallel and a, we have:

m * g * sin(è) = m * (40 m/s)^2 / r

Now, we can cancel the mass (m) on both sides of the equation:

g * sin(è) = (40 m/s)^2 / r

Rearranging the equation to solve for the radius (r), we have:

r = (40 m/s)^2 / (g * sin(è))

Substituting the given values for g (9.8 m/s^2) and è (20°), we can calculate the radius (r).

r = (40 m/s)^2 / (9.8 m/s^2 * sin(20°))

Calculating this expression, we find that the radius (r) is approximately 710.53 meters.

Therefore, to stay on a circular path without depending on friction, the driver should locate his car at a distance of approximately 710.53 meters from the center of the racetrack.