High School Physics Question on Work and Energy

Question

In my textbook, it's written that work only has magnitude and no direction but it is also in negative so is it a vector quantity or scalar?

Answer 1

Work done = change in kinetic energy

1.Scalar quantities can be negative or positive.

2.Change can be negative or positive.

Example-

Throw a ball upward , work done in this case would be negative.(final kinetic energy < initial kinetic energy)

Answer 2

Let me first write the expression for the work done by a force :

If a body moves from position $\mathbf{r}_1$ to position $\mathbf{r}_2$ under the action of force $\mathbf{F}$ then the work done by the force is defined as : $$\int_{\mathbf{r}_a}^{\mathbf{r}_b}\mathbf{F}\cdot d\mathbf{r}=W_{ba}$$

Note: Remember integral used here is a line integral, not an ordinary integral so it depends on the path.

Let's be simple here ( for OP is a High school student)

Consider a 1D definition: If a particle moves a small distance $\Delta \mathbf{x}$ under the action of a force $\mathbf{F}$ then work done by this force will be given by $$\Delta W=\mathbf{F}\cdot \Delta \mathbf{x}$$

remember it's a dot product of force and displacement vector. And as it's dot or scaler product, you get of scalar number which can be positive or negative or even zero.

Naively a vector is a quantity that has direction and magnitude but that's not the here the case. Work has no direction.

The dot product can give a negative value because it's a measure of how much two vectors are aligned with each other. If the vector a more likely in opposite direction, the dot product will be negative. I'll end this with an example of negative work.

Example: Consider a ball that you have projected vertically upward. The displacement vector for a ball is directed upward but the force of gravity is downward. So that results in negative work.

Now, whenever you see negative work done by the force that means force and displacement are in the opposite sense and so the force is actually retarding the motion of the particle. that means the force is resistive in nature for such motion of particles.