KAHIBARO
Table of Contents
Idea of Work for a Constant Force
In physics, work describes how much energy is transferred when a force causes an object to move. In this chapter, the force is constant, which means its magnitude and direction do not change during the motion.
If a constant force acts on an object and the object undergoes a displacement, the work done by the force depends on two things, the size of the force and how much of that force points along the displacement.
Mathematical Definition
For a constant force $\vec{F}$ and a displacement $\vec{d}$, the work done is
$$
W = \vec{F} \cdot \vec{d}
$$
Using the dot product, this becomes
$$
W = Fd\cos\theta
$$
where $F$ is the magnitude of the force, $d$ is the magnitude of the displacement, and $\theta$ is the angle between the force and the displacement.
Important formula:
$$
W = Fd\cos\theta
$$
This formula is valid for a constant force acting during a displacement.
The SI unit of work is the joule, abbreviated J. One joule is equal to one newton times one meter:
$$
1\ \text{J} = 1\ \text{N}\cdot\text{m}
$$
Meaning of the Angle
The angle $\theta$ determines how effective the force is at transferring energy through motion.
If the force points in exactly the same direction as the displacement, then $\theta = 0^\circ$ and
$$
W = Fd
$$
This is the maximum positive work the force can do.
If the force is perpendicular to the displacement, then $\theta = 90^\circ$ and
$$
W = 0
$$
In that case, the force does no work, even though it may still act on the object.
If the force points opposite the displacement, then $\theta = 180^\circ$ and
$$
W = -Fd
$$
This is negative work.
A force does work only through its component along the displacement.
Parallel component: positive work
Perpendicular component: zero work
Opposite component: negative work
Positive, Zero, and Negative Work
Positive work means the force adds energy to the object. A person pushing a box forward while it moves forward is a simple example.
Negative work means the force removes energy from the object. Friction often does negative work because it usually points opposite the motion.
Zero work happens when there is no displacement, or when the force is perpendicular to the displacement.
The table below summarizes these cases.
| Force relative to displacement | Angle $\theta$ | Work |
|---|---|---|
| Same direction | $0^\circ$ | Positive |
| At an angle less than $90^\circ$ | $0^\circ < \theta < 90^\circ$ | Positive |
| Perpendicular | $90^\circ$ | Zero |
| At an angle greater than $90^\circ$ | $90^\circ < \theta \le 180^\circ$ | Negative |
Force Component Along the Motion
Sometimes it is easier to think in terms of components. Only the component of force parallel to the displacement contributes to work.
If the parallel component is
$$
F_{\parallel} = F\cos\theta
$$
then the work is
$$
W = F_{\parallel} d
$$
This form is often the most physically intuitive. It says that only the part of the force that actually helps or opposes the motion matters for work.
Simple Examples
Suppose a person pushes a crate with a constant horizontal force of $50\ \text{N}$ through a horizontal distance of $4.0\ \text{m}$. If the force is in the same direction as the motion, then
$$
W = Fd = (50)(4.0) = 200\ \text{J}
$$
So the person does $200\ \text{J}$ of work on the crate.
Now suppose the same force acts at an angle of $60^\circ$ to the displacement. Then
$$
W = Fd\cos 60^\circ = (50)(4.0)(0.5) = 100\ \text{J}
$$
Only half of the force contributes to the work.
If the force were perpendicular to the motion, then
$$
W = Fd\cos 90^\circ = 0
$$
So no work would be done by that force.
Visualizing Constant Force and Displacement
In this picture, the displacement is horizontal. The angled force has a horizontal component $F\cos\theta$ and a vertical component $F\sin\theta$. Only the horizontal component contributes to the work.
Work by Different Constant Forces
More than one force can act on the same object. Each force can do its own amount of work, depending on its direction relative to the displacement.
For example, if a box moves horizontally across the floor, the applied force may do positive work, friction may do negative work, and the normal force and weight may do zero work if the displacement is horizontal.
| Force | Typical direction | Work for horizontal motion |
|---|---|---|
| Applied push | Along motion | Positive |
| Friction | Opposite motion | Negative |
| Weight | Vertical downward | Zero |
| Normal force | Vertical upward | Zero |
Special Cases
If there is no displacement, then no work is done:
$$
W = 0
$$
even if the force is large.
If the object moves but a particular force is always perpendicular to the displacement, that force does zero work. This idea is very important in circular motion, where some forces can change direction without changing the object's speed through work.
A force alone does not guarantee work.
Work requires both force and displacement.
If either is missing, or if the force is perpendicular to the displacement, then
$$
W = 0
$$
Sign Convention
The sign of work matters.
Positive work means energy is transferred to the object by the force.
Negative work means energy is transferred away from the object by the force.
Zero work means that force does not transfer energy through the displacement considered.
This sign comes directly from $\cos\theta$. Since $\cos\theta$ can be positive, zero, or negative, the work can also be positive, zero, or negative.
Final Summary
For a constant force, work is the dot product of force and displacement:
$$
W = \vec{F}\cdot\vec{d} = Fd\cos\theta
$$
It depends on how much of the force acts along the direction of motion. A force in the direction of motion does positive work, a perpendicular force does zero work, and a force opposite the motion does negative work.
Key result:
$$
W = Fd\cos\theta
$$
Only the component of force parallel to the displacement does work.