Problem 45 - Calculus III

Question

An automobile that weighs \(2700 \mathrm{lb}\) makes a turn on a flat road while traveling at \(56 \mathrm{ft} / \mathrm{sec}\). If the radius of the turn is 70 ft, what is the required frictional force to keep the car from skidding?

Step-by-Step Solution

Answer

The required frictional force is approximately 3757.78 lb.

Step 1: Understand the problem

We need to determine the frictional force required to keep a car from skidding as it makes a turn at a certain speed on a flat road. We are given the weight of the car, the speed, and the radius of the turn.

Step 2: Identify the relevant formula

The frictional force required can be determined using the formula for centripetal force: \[ F_c = \frac{mv^2}{r} \]where \( m \) is the mass of the car, \( v \) is the velocity, and \( r \) is the radius of the turn.

Step 3: Convert weight to mass

The weight of the car is given as 2700 lb. To find the mass \( m \), use the relationship \( W = mg \), where \( g \) is the acceleration due to gravity (32.2 ft/s²). Thus, \[ m = \frac{W}{g} = \frac{2700}{32.2} \approx 83.85 \text{ slugs} \]

Step 4: Plug values into the formula

Substitute the mass \( m = 83.85 \) slugs, velocity \( v = 56 \) ft/s, and radius \( r = 70 \) ft into the centripetal force formula:\[ F_c = \frac{83.85 \times 56^2}{70} \]

Step 5: Calculate the frictional force

Evaluate the expression:\[ F_c = \frac{83.85 \times 3136}{70} = \frac{263044.8}{70} \approx 3757.78 \text{ lb} \]

Key Concepts

Frictional Force in Circular Motion

When a car takes a turn, it navigates a circular path. To avoid skidding, it needs a force to keep it moving in a circle, called the centripetal force. This force in our context is provided by friction between the car's tires and the road surface.

How Friction Works
Frictional force is the force that resists the sliding or rolling of one surface over another. It acts in the opposite direction to the movement, holding the vehicle to the road. The car's tires push against the road to maintain grip, and friction counteracts any sideways movement. This is crucial for negotiating turns.

In this scenario, the frictional force is given by Newton's second law of motion applied to circular motion. It helps convert linear motion into rotational movement without losing contact, essentially keeping the vehicle from skidding off the road. By calculating the centripetal force required for the turn, we determine the exact amount of frictional force needed for safety.

Weight Conversion to Mass

When the weight of the car is given, it is often in pounds (lb). However, for solving physics problems that involve calculations, we often need the mass. Mass is a measure of how much matter the object consists of and is constant regardless of location, while weight is the force exerted by gravity on that mass.

Formula for Conversion
The relationship between weight and mass is defined by the formula: \[ W = mg \]where:

• \( W \) is the weight,
• \( m \) is the mass,
• \( g \) is the acceleration due to gravity.

Converting weight to mass involves rearranging this equation to find \( m \): \[ m = \frac{W}{g} \]The conversion from weight to mass is essential for utilizing equations that require mass. In the original exercise, the car's weight is 2700 lb, and using the acceleration due to gravity (32.2 ft/s² for earth), we convert this to find the mass in units compatible with the other variables, resulting in 83.85 slugs.

Acceleration Due to Gravity

Gravity is a fundamental force that affects any object with mass, pulling it towards the Earth's center. Its acceleration is constant and plays a crucial role in many physics calculations.

Understanding Gravity's Role
The acceleration due to gravity, denoted as \( g \), serves as a bridge between weight and mass. It is the rate at which an object accelerates towards Earth when dropped from a height. On Earth's surface, this acceleration is approximately 32.2 ft/s² in the imperial system.

For physics problems, understanding \( g \) ensures proper unit conversions and accurate calculations of forces. Weight is calculated as the product of mass and gravitational acceleration, hence converting weight to mass requires dividing by \( g \). This understanding is vital for solving any problems related to dynamics and forces in vertical motion scenarios. Whether calculating the centripetal force necessary for a car to stay on track or understanding why objects fall, \( g \) provides the link between terrestrial and dynamic physics principles.