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Centripetal force is the force that keeps an object moving in a circular path. It acts toward the center of the circle. When a car moves on a banked track, centripetal force is crucial because it ensures that the car stays on its circular path without skidding outwards or inwards. The formula for centripetal force is given by:

• \( F_c = \frac{mv^2}{r} \)

where \( F_c \) is the centripetal force, \( m \) is the mass of the object, \( v \) is the velocity, and \( r \) is the radius of the circular path.
On a banked track without friction, gravity and the normal force provide the centripetal force needed. The component of gravitational force along the track substitutes the role of friction, which we'll see in an upcoming section.

The banking angle, often denoted as \( \theta \), is the angle at which the track is inclined relative to the horizontal plane. In our exercise, the banking angle helps generate the needed centripetal force through gravitational components.
This angle is determined by the geometry of the track. To find the banking angle, use the tangent function:

• \( \tan \theta = \frac{\text{height of the wall}}{\text{difference in radii}} \)

For the given problem, using the measurements, we calculated the banking angle \( \theta = \tan^{-1} \left( \frac{18}{165 - 112} \right) \).
This angle ensures that even without friction, the car can move safely on its path at the calculated speeds.

Frictionless motion in the context of a banked track implies that the car relies solely on the inclination of the track and gravity to maintain its path. Here, the normal force and the gravitational force's vertical component align to create the centripetal force.
For frictionless travel, the speed must match the theoretical speed calculated using the formula:

• \( v = \sqrt{r \cdot g \cdot \tan \theta} \)

where \( v \) is the speed, \( r \) is the radius, \( g \) is the gravitational acceleration, and \( \tan \theta \) is from the banking angle.
Achieving this velocity means that the car can safely navigate the track without relying on any additional frictional support.

In circular motion, the radius is the distance from the center of the circle to any point on its circumference. It plays a significant role in determining the needed centripetal force and, subsequently, the velocity at which a car can safely travel on a banked track.
For our scenario, there are two radii: 112 m (smallest path) and 165 m (largest path). These radii represent different paths a car can take on the track.
The value of the radius directly impacts the velocity calculation. A larger radius results in a higher speed permissible by the car, as seen in the calculations for both paths using the centripetal force formula provided.

Gravitational acceleration is a constant force acting on objects due to Earth's gravity. Denoted by \( g \), its standard value is \( 9.81 \ \text{m/s}^2 \). This value impacts how forces work on the banked track, providing the component of centripetal force needed in frictionless conditions.
Gravity not only pulls objects downward but also plays a key role in allowing the car to maintain its circular path on the banked track. When calculating speed for frictionless motion, \( g \) is used alongside the radius and banking angle in the formula:

• \( v = \sqrt{r \cdot g \cdot \tan \theta} \)

This demonstrates how gravitational acceleration contributes to the speed at which the car can safely travel.