91Ó°ÊÓ

Static friction is the force that keeps an object at rest when other forces try to move it. It acts between two surfaces, preventing them from sliding past each other. In this exercise, static friction plays a crucial role in allowing the race car to navigate the banked curve without skidding.

This frictional force depends on two factors:

• The normal force (perpendicular force) exerted by the road on the tires.
• The coefficient of static friction, which is a measure of the stickiness between the two surfaces.

The maximum static frictional force can be calculated using the formula: \[ f_s = ext{coefficient of static friction} imes ext{normal force} \]This means it's important to know the normal force, as it directly influences the frictional force that prevents skidding.

When cornering, if the centripetal force, required to keep the car moving in a circle, exceeds this maximum static frictional force, the car will begin to skid. However, as long as the frictional force is greater than or equal to the required centripetal force, the car remains securely on the track.

Centripetal acceleration is crucial for understanding how objects move in circular paths. It is the acceleration directed towards the center of the circle, keeping the object moving along the curve. For this exercise, the race car experiences centripetal acceleration as it rounds the track.

The formula for centripetal acceleration is:\[ a_c = \frac{v^2}{r} \]where:

• \( v \) is the velocity of the car.
• \( r \) is the radius of the circular path.

Centripetal acceleration is always perpendicular to the velocity of the car because it's directed towards the center of the circle. This is a key requirement for maintaining circular motion.

In the context of this exercise, the balance of forces involves static friction and gravitational forces contributing toward generating enough centripetal acceleration to keep the car on track without skidding. The car’s speed determines how effectively this balance is achieved.

The normal force is a force exerted by a surface in response to an object resting on it. It acts perpendicular (or "normal") to the surface, which, in this scenario, is the banked curve of the racetrack.

On a flat surface, normal force usually equals the gravitational force exerted by the object (weight). However, on an inclined surface, such as the banked curve described, the normal force changes its magnitude because it's influenced by both gravity and the angle of the incline.

For a banked curve, the normal force is calculated as:\[ F_N = mg\cos(\theta) \]Here,

• \( m \) is the mass of the race car.
• \( g \) is the acceleration due to gravity (approximately \(9.8 \text{m/s}^2\)).
• \( \theta \) is the angle of the bank.

In this exercise, after computing this, the normal force acts as a stabilizing component, helping the car to effectively use static friction to maintain its trajectory and accommodate the necessary centripetal force for staying on the curved path.