Problem 34 - Physics for Scientists and Engineers with Modern Physics

Question

(I) What is the maximum speed with which a 1200 -kg car can round a turn of radius 80.0 \(\mathrm{m}\) on a flat road if the coeffi- cient of friction between tires and road is 0.65\(?\) Is this result independent of the mass of the car?

Step-by-Step Solution

Answer

The maximum speed is approximately 22.57 m/s, and it is independent of the mass of the car.

Step 1: Understand the Problem

We need to find the maximum speed of a car that can safely navigate a curve without skidding. This is determined by the frictional force between the tires and the road, which provides the necessary centripetal force. The formula for frictional force is \( f = \mu \cdot N \), where \( \mu \) is the coefficient of friction and \( N \) is the normal force.

Step 2: Set Up the Formula for Centripetal Force

The centripetal force needed to keep the car moving in a circle is given by \( F_c = \frac{m \cdot v^2}{r} \). Here, \( m \) is the mass of the car, \( v \) is the speed, and \( r \) is the radius of the turn.

Step 3: Relate Frictional and Centripetal Forces

For the car not to skid, the maximum frictional force must be equal to the required centripetal force. Therefore, set \( \mu \cdot m \cdot g = \frac{m \cdot v^2}{r} \). Notice that the mass \( m \) cancels out from both sides of the equation.

Step 4: Solve for Maximum Speed \( v \)

Rearrange the equation to solve for the speed \( v \):\[\mu \cdot g = \frac{v^2}{r} \implies v^2 = \mu \cdot g \cdot r \implies v = \sqrt{\mu \cdot g \cdot r}\]Insert the values: \( \mu = 0.65 \), \( g = 9.8 \, \mathrm{m/s^2} \), and \( r = 80 \, \mathrm{m} \).

Step 5: Calculate the Numerical Value

Substitute the known values into the equation:\(v = \sqrt{0.65 \times 9.8 \times 80} = \sqrt{509.6} \approx 22.57 \, \mathrm{m/s}\)

Step 6: Examine Dependence on Mass

Since the mass cancels out from both sides of the equation during the derivation, the maximum speed does not depend on the mass of the car.

Key Concepts

Centripetal Force

Centripetal force is a fundamental concept in circular motion. It refers to the force required to keep an object moving in a circular path. Imagine whirling a stone tied to a string; the tension in the string keeps the stone from flying away. That tension acts as the centripetal force. In mathematical terms, centripetal force is expressed as:

• \(F_c = \frac{m \cdot v^2}{r}\)

Where:

• \(F_c\) is the centripetal force
• \(m\) is the mass of the object
• \(v\) is the velocity or speed of the object
• \(r\) is the radius of the circular path

In the scenario of a moving car taking a curve, the centripetal force is what keeps the car from sliding outward. It's essential for understanding how and why the car can safely navigate the turn without veering off the road. The force must be directed towards the center of the circular path, preventing the car from spiraling out of the curve.

Frictional Force

Frictional force is the resistance force that occurs when two surfaces interact. It plays a crucial role in the dynamics of motion, especially in scenarios such as a car rounding a turn. This force opposes the motion of the car's tires against the road. Without friction, cars would be unable to turn effectively or come to a stop.The frictional force can be calculated using:

• \(f = \mu \cdot N\)

Where:

• \(f\) is the frictional force
• \(\mu\) is the coefficient of friction
• \(N\) is the normal force, which typically equals \(m \cdot g\) where \(g\) is the acceleration due to gravity

In vehicle dynamics, the available frictional force between the tires and the road is vital for providing the required centripetal force. If the car is moving too fast, and the frictional force is inadequate, the car will skid. Thus, balancing frictional force is key to making a safe turn.

Coefficient of Friction

The coefficient of friction is a measure that describes how much frictional force is present between two surfaces. It is a unitless value, representing the ratio of the force of friction between two bodies and the force pressing them together. In simpler terms, it tells us how easily one object will slide over another.In the context of a car moving on a road, the coefficient of friction (\(\mu\)) is crucial in determining the grip of the tires:- A higher coefficient indicates greater friction, allowing higher speed during turns without slipping.- A lower coefficient signals less grip, leading to potential slipping at lower speeds.In our example, with a coefficient of friction of 0.65 on a flat road, we find that this value signifies moderate to strong grip. It permits the car to make turns at certain speeds without skidding. Understanding this coefficient helps predict whether the tires will maintain traction or lose it, which is especially important for safe driving practices in different weather conditions.