Integrals and Work: Introduction

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mes4.4 K5 years agoSteemit4 min read

In this video I go over the concept of work in terms of the definition of work as the force multiplied by displacement (W = F·d) as well as defining it using integrals as the area under the curve of a force, f(x), function where x is the position. I also go over the units of work which are Joules (newton-meters) in SI Units and in Foot-Pounds in US Customary units.

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Integrals and Work

The term work is used in everyday language to mean the total amount of effort required to perform a task. In physics it has a technical meaning that depends on the idea of a force.

Intuitively, you can think of a force as describing a push or pull on an object - for example, a horizontal push of a book across a table or the downward pull of Earth's gravity on a ball.

In general, if an object moves along a straight line with position function s(t), then force F on the object (in the same direction) is defined by Newton's Second Law of Motion as the product of its mass m and its acceleration:

In the SI metric system, the mass is measured in kilograms (kg), the displacement in meters (m), the time in seconds (s), and the force in newtons (N = kg·m/s²).

Thus, a force of 1 N acting on a mass of 1 kg produces an acceleration of 1 m/s².

In the U.S. Customary system the unit of force is the pound or lb.

In the case of constant acceleration, the force F is also constant and the work done is defined to be the product of the force F and the distance d that the object moves:

work = force X distance

W = F·d

If F is measured in newtons and d in meters, then the unit for W is a newton-meter (N-m), which is called a joule (J).

If F is measured in pounds and d in feet, then the unit for W is a foot-pound (ft-lb), which is about 1.36 J.

The above work equation is defined for when force is constant. But what happens if the force is variable?

Let's suppose that the object moves along the x-axis in the positive direction from x = a to x = b and at each point x between a and b a force f(x) is acted upon the object, where f is a continuous function.