The curves on a race track are banked to make it easier for cars to go around the curves at high speed. Draw a free-body diagram of a car on a banked curve. From the motion diagram, find the direction of the acceleration

(a) What exerts the force in the direction of the acceleration?
(b) Can you have such a force without friction? (Explain.)

To draw a free-body diagram of a car on a banked curve, we need to consider the forces acting on the car.

Let's assume that the car is moving towards the right on the banked curve. Here are the forces involved:

1. Weight (mg): This force acts vertically downwards, pulling the car towards the center of the Earth.

2. Normal force (N): This force is perpendicular to the surface of the banked curve and acts perpendicularly away from the surface of the banked curve. It counteracts the vertical component of the weight.

3. Friction force (f): This force acts horizontally towards the center of the curve, parallel to the surface of the banked curve. It helps provide the necessary centripetal force to keep the car moving in a curved path.

Now, let's focus on the direction of acceleration. Since the car is moving in a curved path on the banked curve, its acceleration has both a radial component (directed towards the center of the curve) and a tangential component (directed along the direction of motion).

(a) The force exerted in the direction of the acceleration is the horizontal component of the normal force (N). This force is responsible for providing the necessary centripetal force that helps the car turn around the curved track.

(b) No, it is not possible to have a force in the direction of the acceleration without friction. Friction is required to provide the centripetal force because, without it, the car would not be able to maintain its circular path. Friction between the tires of the car and the surface of the banked curve is necessary to prevent the car from sliding off the curve.

To draw a free-body diagram of a car on a banked curve, we consider the forces acting on the car. The forces involved are the normal force (N), the gravitational force (mg), and the friction force (f).

Here's the free-body diagram:

N
|\
| \
f<-----|__\----->

mg

The normal force (N) acts perpendicular to the surface at a right angle. The gravitational force (mg) acts vertically downward towards the center of the Earth. The friction force (f) acts parallel to the surface of the track.

From the motion diagram, we can find the direction of the acceleration. The acceleration points towards the center of the circular motion and is directed along the face of the banking. Therefore, the direction of the acceleration is towards the center of the circular track.

Now, let's answer the specific questions:

(a) What exerts the force in the direction of the acceleration?
The component of the normal force (Nsinθ) exerts the force in the direction of the acceleration. It helps to provide the centripetal force required to maintain the car's circular motion.

(b) Can you have such a force without friction? (Explain.)
No, such a force cannot exist without friction. The friction force (f) acts parallel to the surface of the track and provides the necessary inward force to counterbalance the centrifugal force (the apparent force that pushes objects away from the center of the circular motion). Without friction, the car would slide off the track in a straight line instead of following the curved path. Therefore, friction is crucial for the car to maintain traction and safely negotiate the banked curve at high speeds.