# Why can work done by friction be negative if work is a scalar?

Work done by an object can be defined as the force times the distance traveled in the direction of the force. I've read from the internet that the frictional force acting on an object sliding over a surface can be negative since friction's force is in the opposite direction as the objects distance. However, distance (being a scalar) is not displacement, hence the direction of frictional force shouldn't determine the negative sign?

In addition, since work is a scalar quantity, how can it be negative?

In physics, work is defined as the energy transferred to or from an object by means of a net force acting on the object. If energy is transferred to the object, the work done by the net force is positive. If energy is transferred from the object, the work done by the net force is negative.

The work $W$ done by a constant force $\vec{F}$ is given by

$$W = \vec{F} \cdot \Delta\vec{x},$$

where $\Delta\vec{x}$ is the object's displacement while the force is acting on it. In this expression, $\vec{F}$ can be the net force (giving the total work done on the object) or it can be one of the individual forces (giving the work done by that force-- adding up all the individual work values gives the total work done by the net force).

From the definition of the dot product, we can see that $\vec{F}$ does positive work when it has a component in the same direction as $\Delta\vec{x}$ and it does negative work when it has a component in the opposite direction as $\Delta\vec{x}$.

It is possible to prove that the total work $W$ done on an object is equal to the object's change in kinetic energy $\Delta{K}$ over the time the work was being done:

$$W = \Delta{K}.$$

If a frictional force $\vec{F}$ acting on a sliding object is the only force doing work (think of an object in a vacuum sliding over a rough, horizontal surface), then we will calculate a negative value for $W$ for two reasons. First, $\vec{F}$ has a component opposite the direction of the displacement $\Delta\vec{x}$. Second, the object will slow down, losing kinetic energy, so that $\Delta{K}$ is negative.

frictional force acting on an object sliding over a surface can be negative since friction's force is in the opposite direction as the objects distance.

The word distance (a scalar) in this definition should be replaced by the word displacement (a vector). The dot (scalar) product of two vectors, force and displacement, yields work (a scalar).

In addition, since work is a scalar quantity, how can it be negative?

Sure, as many other scalars, like distance or potential energy. In many cases, it is just a matter of convention.

• Distance can be negative? Can you give an example? Aug 29 '18 at 16:37
• @M.Enns You can choose a direction from any point along a curve, for instance, and consider the distance from that point in one direction positive and in the other negative. A proper term of this is a directed distance. Agree that this is not the best example, but it does not change my point that scalars could be negative.
– V.F.
Aug 29 '18 at 17:06

Yes work done is a scalar and it does not have a direction. But in a sense, it does have a scale to it. So it can be negative/positive.

Example, the most common is temperature. It can have negative and positive values.

A more common use of negatives in work done is also electrical potential energy/electrical work.

When u have a force in the opposite direction, hence W= -F • D. So the scalar quantity is negative. Just that, that negative sign isnt the direction per se but the scale to the value. Wat does a vector really mean?

Yes it has a magnitude and a direction. But wat does it mean to have a direction?

If u really think abt it, all vectors are related to motion. So when ur moving through space, u have direction. So if its related to motion, then its most likely a vector.

Mathematically to ensure that its a vector, either it must be a a cross product of 2 vectors or addition of 2 vectors.

For scalars, if u think again, all scalars are just intrinsic values of measurement. Like mass, temperature, energy/work, power.

Scalars can be negative. The word scalar refers to scaling up or down. With a negative scalar you "flip" the map. It is a mathematical definition. No issue with that.

You might be thinking about the term magnitude which indeed cannot be negative by definition.

Regardless, you are essentially asking how to understand the sign of the work property. Remember that the sign of any quantity in physics is nothing but a mathematical invention we have made to indicate "direction" in various scenarios.

• A negative force force has no physical meaning, but just tells us that the force points opposite to whichever direction we've chosen to be positive.
• Regarding energy flow, a "direction" could be interpreted as energy that enters (is absorbed by) or leaves (is spent by) an object or system. If you've chosen "absorbed by" as positive, then negative work just tells you that this amount of energy is removed from the object/system.

With this definition of signs in relation to work, we can invent mathematical formulas where the sign of the result fits the definition. Thus, work has been defined as

$$W=\vec F\cdot \vec s$$

where $$\vec s$$ is displacement. These are vector quantities, and since the dot product becomes negative for oppositely pointing vectors, we can define work as the mechanical energy supplied to a object by a force that pushes the system through the displacement $$\vec s$$.

Since friction pushes against the displacement, if does not supply energy to the system but rather removes energy from the system. Thus it should be negative. And that fits with the dot product formula.