7.1 Work: The Scientific Definition

Work: The Scientific Definition

In physics, work has a precise meaning: it measures energy transferred to or from an object by a force acting over a distance. This concept connects force and motion to energy, which makes it foundational for everything else in this unit.

Work Calculation for Applied Forces

Work equals the force applied to an object multiplied by the displacement in the direction of that force:

$W = Fd$

where $W$ is work, $F$ is the magnitude of the force, and $d$ is the displacement.

The unit of work is the joule (J), where $1 \text{ J} = 1 \text{ N·m}$. So if you push a crate with a 10 N force and it moves 5 m in the direction you're pushing, the work done is:

$W = (10 \text{ N})(5 \text{ m}) = 50 \text{ J}$

A few properties worth knowing:

• Work is a scalar quantity. Unlike force (a vector with magnitude and direction), work only has magnitude. It can be positive, negative, or zero, but it doesn't point in a direction.
• Work represents a transfer of energy. When you do positive work on an object, you're adding energy to it.
• Net work is the sum of work done by all individual forces acting on an object: $W_{net} = W_1 + W_2 + ... + W_n$

Sign of Work in Force-Displacement Scenarios

The sign of work tells you whether energy is being transferred to or from an object. It depends on the angle between the force vector and the displacement vector.

• Positive work (angle less than 90°): Force and displacement point in the same general direction. Energy is added to the object.
• Negative work (angle greater than 90°): Force and displacement point in generally opposite directions. Energy is removed from the object.
• Zero work (angle exactly 90°): Force is perpendicular to displacement. No energy is transferred.

Here are concrete examples for each case:

Positive work:

• Pushing a box forward across the floor. Your push and the box's motion point the same way.
• A person walking uphill, with their applied force and displacement both directed along the incline.

Negative work:

• Slowly lowering a box to the ground. Your hands exert an upward force, but the box displaces downward.
• Friction on a sliding object. The friction force always opposes the direction of motion.

Zero work:

• Carrying a box horizontally at constant speed. The force you exert is upward (supporting its weight), but the displacement is horizontal. Those are perpendicular, so the work you do on the box is zero.
• Gravity acting on an orbiting satellite. The gravitational force points toward Earth's center, but the satellite's displacement at any instant is perpendicular to that.

Application of the Work Formula

Most real-world forces don't line up perfectly with the direction of motion. The general work formula accounts for this using the angle $\theta$ between the force and displacement vectors:

$W = Fd\cos\theta$

Notice how this formula contains the simpler cases:

• When force and displacement align ($\theta = 0°$), $\cos 0° = 1$, so $W = Fd$.
• When they're perpendicular ($\theta = 90°$), $\cos 90° = 0$, so $W = 0$.
• When they're opposite ($\theta = 180°$), $\cos 180° = -1$, so $W = -Fd$.

To solve a work problem using this formula:

1. Identify the force $F$, the displacement $d$, and the angle $\theta$ between them.
2. Plug the values into $W = Fd\cos\theta$.
3. Calculate, paying attention to the sign of your answer.

Example 1: A person pulls a sled with a 50 N force at 30° above the horizontal for 10 m. The angle between the force and the horizontal displacement is 30°.

$W = (50 \text{ N})(10 \text{ m})\cos(30°) = 500 \times 0.866 \approx 433 \text{ J}$

Only the horizontal component of the pulling force does work here, since the sled moves horizontally.

Example 2: A force of 30 N is applied parallel to an incline surface, and a block moves 2 m along that incline. Since the force is parallel to the displacement, $\theta = 0°$, and the work is simply:

$W = (30 \text{ N})(2 \text{ m})\cos(0°) = 60 \text{ J}$

Energy and Work

Work connects directly to changes in mechanical energy (kinetic energy and potential energy). When net work is done on an object, its kinetic energy changes by that amount. This relationship, called the work-energy theorem, will be explored in upcoming sections.

The conservation of energy principle governs these relationships: energy isn't created or destroyed, only transferred or converted. Power, which you'll also encounter soon, measures the rate at which work is done, in units of watts ($1 \text{ W} = 1 \text{ J/s}$).