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# Work

## Work definition

Work $(W)$ is done by a force when a force moves an object.

More precisely, for work to be done, a force must be applied to an object and the object must move in the direction of the force.

\begin{align*}\Tblue{\text{work done}} =& \Tred{\text{force}} \times \Tgreen{\text{distance moved in direction of force}} \\ \Tblue{W}=&\Tred{F}\times\Tgreen{d}\end{align*}

• If you lift a book from a table you apply a force on the book and it moves in the direction of the force. Work is done.
• If you push against a wall, you exert a force on the wall but no movement occurs. There is no work done.
• A ball continues to move after it has been kicked. It is still moving but there is no longer any force acting on it. No work is being done.

Work is a scalar quantity (it has no direction).

The definition in physics contrasts with the use in everyday English. Any sort of task undertaken for a salary is considered to be work.

A crane does work by lifting objects like this container.

## Unit of work

The SI unit of work is the joule $(\text{J})$. One joule is the work done when a force of $1\text{ N}$ displaces an object by a distance of $1\text{ m}$ in the direction of the force.

In SI base units, the joule can be expressed as $\text{N m}$. That is also the unit of the moment (or turning effect) of a force but moments and and work are very different and should not be confused.

Work done is a scalar quantity and a moment is a vector quantity. Work is always expressed in $\text{J}$ and moments in $\text{N m}$.

The quantities weight and work both have the same symbol ($W$), so it is important not to confuse the two in equations.

The joule is named after the scientist James Prescott Joule. Joule spent a lot of time working on work. He also had an impressive beard.

## Positive and negative work

When the object moves in the same direction as the force, the work done is positive (i.e. work is done by the force).

When the object moves in the opposite direction of the force, the work done is considered to be negative (i.e. work is done by the object).

When a man tries to push a car, but the car moves further towards the man, the man performs negative work on the car.

When positive work is done by the force, negative work is done by the object and vice versa. This arises from Newton's third law (for every action, there is an equal and opposite reaction).

When work is said to be done by the object, it is in fact the reaction force of the object that does work. This reaction force is necessarily in the opposite direction of the applied force and hence takes on a negative sign.

The work done by the object is therefore the negative of the work done by the force.

If the motion of an object is perpendicular to the direction of the force, no work is done by the force.

## Implications of Newton's second law for work done

Newton's second law $F=ma$ (where $F$ is the net force) implies that:

• Work is done on an object by a particular force regardless of whether the object accelerates. The only requirement for work to be done is that the object moves in any direction (except perpendicularly) relative to the force.

• Work is done on an object by the resultant force if the object accelerates and does not move perpendicularly to the resultant force.

A force that maintains an object at constant speed against friction does positive work. However, the net work done by the force and friction is zero (as the net force is zero).

## Work done by forces at an angle

There is no work done by a force when an object does not move or moves in a direction which is perpendicular to the direction of the force.

The force which acts on an object that moves in circular motion does no work as the force is always perpendicular to the motion of the object.

If the direction of the force is at an angle $\theta$ to the displacement of the object, the work done on the object is the product of the component of the force along the direction of motion of the object (i.e. the direction of the displacement $\vecphy{s}$ of the object) and the magnitude of displacement: $$W=Fs\cos\theta$$

Note that $\theta=0$ corresponds to the force pointing in the same direction as the displacement and $\theta=\pi$ indicating that the force and displacement point in opposite directions.

$F=$force; $s=$displacement in the direction of the force.

## Force-displacement graph and work done by gases

If the force varies in magnitude with the displacement of the object, the total work done by the force is given by the area under the graph of $F$ vs $s$.

$F=$force; $s=$displacement in the direction of the force.

The expansion or compression of a gas acted on by an external pressure also involves work.

This is because a gas exerts a force on the inner surfaces of a container due to pressure. When the volume of the container changes, work is done by the gas (or the container).