Question

What is the net force acting on a 3900 -lb automobile accelerating at \(13 \mathrm{ft} / \mathrm{s}^{2}\) ?

Short answer

The net force acting on the car is calculated as the product of the car's mass and its acceleration.

Step-by-step solution

Convert weight to mass

The weight of an object is the force with which it is pulled towards the center of the Earth by gravity. It is related to the object's mass by the equation \( Weight = Mass \times Gravity \). In US customary units, the value of gravity is approximatively 32.2 ft/s². So, to calculate the mass of the car in slugs (the US customary unit for mass), one can rearrange the equation and divide the weight by the gravitational acceleration: \( Mass = \frac{Weight}{Gravity} = \frac{3900 lb}{32.2 ft/s^2} \).

Apply Newton’s second law

Newton's second law of motion, \( Force = mass \times acceleration \), can be used to find the net force acting on the car. The mass of the car has been found in step 1 and the acceleration is given as 13 ft/s². The force can be found by multiplying these two values together: \( Force = mass \times acceleration \).

Calculation and Result

Now, plug in the values obtained for mass and the given acceleration into the formula from step 2 to find the net force. This will give the solution to the problem.

Key concepts

Newton's Second Law

Newton's Second Law of Motion is a fundamental principle that describes the relationship between an object's mass, its acceleration, and the net force applied to it. When you apply a force to an object, it accelerates, and this acceleration is directly proportional to the force applied and inversely proportional to the mass of the object.

The law is succinctly expressed in the formula \( F = ma \), where \( F \) is the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration. This principle is crucial in solving problems involving motion, enabling us to predict how objects will move under various forces.

When using this law, it's essential to consider all forces acting on the object and calculate the net force, which is the vector sum of all individual forces. Understanding this law is key to explaining and predicting physical phenomena in classical mechanics.

Converting Weight to Mass

In physics problems, it's often necessary to convert weight to mass. Weight is a measure of the gravitational force on an object, while mass is the amount of matter in the object and does not change with location, even when gravity varies.

To convert weight (measured in pounds in the US customary units) to mass (measured in slugs in the US), we use the formula \( m = \frac{W}{g} \), where \( m \) is mass, \( W \) is weight, and \( g \) is the acceleration due to gravity, approximately \( 32.2 \text{ ft/s}^2 \) on Earth. By dividing weight by gravitational acceleration, you obtain the mass, which can be used in further calculations, such as applying Newton's second law.

Acceleration

Acceleration is a measure of how quickly an object's velocity changes. It's defined as the rate of change of velocity with time and can be calculated as \( a = \frac{\Delta v}{\Delta t} \), where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time.

Acceleration can be a result of changes in speed, direction, or both. It's a vector quantity, which means it has both magnitude and direction. In everyday language, we often use 'acceleration' to describe something speeding up, but in physics, it can also refer to slowing down (deceleration) and changing direction. For the motion of automobiles and other objects, understanding acceleration is necessary to predict and analyze their behavior under various forces.

US Customary Units

The United States uses a system of measurement called US customary units, which are different from the metric system commonly used in other parts of the world. In the context of physics and Newton's laws, important US customary units include pounds (lb) for force or weight, feet per second squared (ft/s²) for acceleration, and slugs for mass.

Understanding and converting between these units is vital in solving physics problems. For instance, when calculating the net force on an object, such as a car, its weight must be given in pounds, and the acceleration must be in feet per second squared. To connect weight and mass, we use the gravitational constant, \( 32.2 \text{ ft/s}^2 \), as part of the conversion process. Mastery of these units and conversions is essential for accuracy in calculations.