91Ó°ÊÓ

Centripetal force is essential for keeping an object moving in a circular path. Imagine a car driving around a curve on the road. The centripetal force is what prevents it from skidding outwards due to its inertia. This force is directed towards the center of the circle, ensuring that the car maintains its circular motion.

The formula for calculating centripetal force is:

• \( F_c = \frac{mv^2}{r} \)

where \( m \) is the mass of the car, \( v \) is its velocity, and \( r \) is the radius of the curve.

In this exercise, a banked curve is designed to provide centripetal force without relying on friction. Understanding how centripetal force works helps in setting up the necessary conditions for safe and efficient turning on curves.

A banked curve is a roadway or track that is raised on the outer edge compared to the inner edge. The purpose of this design is to allow vehicles to navigate the curve at higher speeds without the risk of skidding, even in the absence of friction.

When a car takes a banked curve, the design of the road provides the necessary centripetal force through the component of the normal force. This ensures that the car stays on its path despite the curve. The banking angle plays a critical role in achieving this by creating an appropriate path for the force vector.

• In a banked curve, the horizontal component of the normal force acts as the centripetal force.
• The angle of banking reduces the reliance on friction.

Banked curves are common in highways and racing tracks, where maintaining speed and safety are priorities.

A free-body diagram is a visual representation that helps in isolating the forces acting on an object. In the scenario of a car navigating a banked curve, it's vital to understand these forces to determine how the car can take the turn without skidding.

Here are the main forces represented in the free-body diagram for a car on a banked curve:

• Gravitational force (\( mg \)), acting vertically downward.
• Normal force (\( N \)), acting perpendicular to the surface of the road.
• Centripetal force component, coming from the horizontal component of the normal force.

Drawing a free-body diagram aids in analyzing these forces and understanding how they interact to achieve equilibrium and maintain motion on the banked turn. This visualization helps to formulate the required equations for solving physics problems.

The calculation of the banking angle is key to ensuring vehicles can safely take a curve without relying on friction. The angle determines how much the road should be inclined so that the centripetal force required for navigating the turn comes completely from the road design and not from external factors like tire friction.

The relationship for calculating the banking angle \( \theta \) is:

• \( \tan\theta = \frac{v^2}{rg} \)

where \( v \) is the velocity of the vehicle, \( r \) is the radius of the curve, and \( g \) is the acceleration due to gravity.

In the given exercise, converting the vehicle's speed and using this formula allows the calculation of the exact angle. By substituting the appropriate values, we find that \( \theta = 12.77^\circ \). This calculation ensures the car can travel safely along the banked curve at the designated speed.