Problem 4 - College Physics

Question

A flat (unbanked) curve on a highway has a radius of 220 \(\mathrm{m}\) .A car rounds the curve at a speed of 25.0 \(\mathrm{m} / \mathrm{s}\) . (a) Make a free body diagram of the car as it rounds this curve. (b) What is the minimum coefficient of friction that will prevent sliding?

Step-by-Step Solution

Answer

The minimum coefficient of friction \( \mu \) is approximately 0.291.

Step 1: Understand the Forces

As the car travels around the curve, it is subject to two main forces: the gravitational force acting downwards, and the frictional force that provides the necessary centripetal force to keep the car moving in a circular path. There is no normal force component aiding centripetal force on a flat surface.

Step 2: Draw the Free Body Diagram

In the free body diagram, illustrate the car as a box. Draw an arrow downwards representing the gravitational force \( F_g = mg \), and an arrow to the left (in the direction towards the center of the curve) for the frictional force \( F_{f} \), which acts as the centripetal force.

Step 3: Determine the Required Frictional Force

For the car to move in a circle, the frictional force must equal the centripetal force required. Using the formula for centripetal force \( F_c = \frac{mv^2}{r} \), where \( m \) is the car's mass, \( v = 25.0 \mathrm{m/s} \) is the speed, and \( r = 220 \mathrm{m} \) is the radius, we calculate the force needed for circular motion.

Step 4: Relate Frictional Force and Normal Force

The frictional force \( F_{f} \) is given by \( F_{f} = \mu F_n \), where \( \mu \) is the coefficient of friction and \( F_n \) is the normal force. For a flat surface, the normal force \( F_n = mg \). We need to ensure that \( \mu mg \geq\frac{mv^2}{r} \).

Step 5: Solve for Minimum Coefficient of Friction

Cancel out the mass \( m \) from both sides of the inequality: \( \mu g \geq \frac{v^2}{r} \). Rearrange to find \( \mu \):\[ \mu \geq \frac{v^2}{gr} \].Substitute the values: \( v = 25.0 \mathrm{m/s}, g = 9.8 \mathrm{m/s^2}, r = 220 \mathrm{m} \).\[ \mu \geq \frac{(25.0)^2}{9.8 \times 220} \approx 0.291. \]

Key Concepts

Frictional Force

Frictional force plays a crucial role when a vehicle negotiates a curve on a flat (unbanked) road. This force opposes the natural tendency of an object to move in a straight line and instead helps it to follow the curved path. When a car rounds a curve, the frictional force acts towards the center of the circle formed by the curve. This is why it is also known as the centripetal force in this context.
The magnitude of the frictional force depends on the nature of the surfaces in contact and their relative motion. For a car on a road, this means the type of tires and the condition of the road surface determine the friction level. The frictional force can be calculated using the equation:

• Frictional force, \( F_{f} \) = \( \, \mu \, F_{n} \)

where \( \mu \) is the coefficient of friction and \( F_{n} \) is the normal force. On a flat road, the normal force \( F_{n} \) equals the gravitational force \( mg \), the weight of the car, directed upward, balancing the weight of the car.

Coefficient of Friction

The coefficient of friction \( \mu \) is a dimensionless value that describes how much friction is available between two surfaces. In our context, it tells us how much grip the road provides to the car tires. A higher coefficient means more friction, which in turn means the car can handle tighter curves at higher speeds without sliding.
To prevent sliding while rounding a curve, the frictional force must be at least equal to the centripetal force needed. This creates the necessity to compute a minimum coefficient of friction using the formula:

• \( \mu \geq \frac{v^2}{gr} \)

where \( v \) is the car's speed, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the curve. Substituting in the given values allows us to determine the least amount of friction necessary: \( \mu \geq 0.291 \). This means a coefficient of friction of at least 0.291 is required to ensure the car does not slide off the road.

Free Body Diagram

A free body diagram is an essential tool in physics that helps in understanding and analyzing forces acting on an object. It's a simple pictorial representation where forces are depicted as arrows starting from a point or a box representing the object.
For a car moving around a curve, the free body diagram will include:

• A downward arrow representing gravitational force \( F_{g} = mg \), showing the weight of the car.
• An arrow directed towards the center of the curvature representing the frictional force \( F_{f} \), acting as the centripetal force.

These forces add clarity to the problem, allowing us to evaluate and ensure the frictional force is sufficient to maintain the car's curved path. By analyzing this diagram, we can see how friction is essential in providing the needed centripetal force to avoid skidding.