# Is Moment of force equal to work done by force and equal to gravitational potential energy? (M = W = g.p.e. )

Ridiculous question but... If these equations are true:

Moment of force = force * perpendicular distance from pivot to force (M=Fd)

And

Work done by force = force * distance moved by force in the direction of force ( W = Fd)

Does that mean...

Moment of force = work done by force

(I am aware that the distances here are not the same, one is when it comes to pivots and the other for work)

But both have the same units, Nm (Newton * meter)

And

Force and distance in both equations are indirectly proportional, with either directly proportional with their multiplied value..:

If distance is contant and force increases, moment of force increases..

If force is contant and distance decreases, work done by works decreases..

Additionally, Newton * meter is Joules..in case of work.. (Nm = J)

Does this mean..

Moment of force = work done by force ?

BUT IT DOESN'T STOP THERE.

If weight = mass * acceleration due to gravity = force (W = mg = F)

And

gravitational potential energy = weight * height = mass * g * height = force * height = MOMENT OF FORCE = WORK DONE BY FORCE. (g.p.e. = mgh = Fh = Fd = M = W )

Concluding that...

Moment of force = work done by force = gravitational potential energy (M = W = g.p.e.)

--I find this ridiculous but I've been quite disturbed by the fact that this hasn't been discussed and answered before (as far as I've seen)

First of all I suggest you take a look on how to use MathJax for correct formatting of the question as it is a bit difficult to read.

Now to the physics. Actually the torque of a force is defined as $$\vec{M}=\vec{r}\times \vec{F}$$ where $$\vec{r}$$ is the distance from a certain point to the point where the force is exerted. Work is defined as $$W=\int \vec{F}\cdot d\vec{r}$$ where $$d\vec{r}$$ defines a trajectory in space. Even though they have the same units ($$\mbox{force} \times \mbox{distance}$$), they do not represent the same physical quantities. You have to notice that torque is defined with respect to one point, which can be seen in the next image (sorry for the dimensions)

On the other hand, work needs a path to be defined, a path through which the force $$\vec{F}$$ is exerted. As it is an integral, it can be interpreted as the area under a curve:

Another very important difference is that work is always an scalar quantity, i.e. a number, whereas torque is a vector. The only "equivalence" you can find in your reasoning is between work and energy, which really represent the same physical quantity, as stated in the first law of thermodynamics. I hope this solves your question.

Does that mean...

Moment of force = work done by force

No.

In the case of angular motion (motion produced by torque, or as you say, moment) force is replaced by torque and linear displacement is replaced by angular displacement. Or,

$$W=\int \overrightarrow F.d\overrightarrow x$$

Is replaced by

$$W=\int \overrightarrow τ.d\overrightarrowθ$$

Where $$θ$$ is the angular displacement in radians as a result of applying the torque.

Torque and force are related by

$$\overrightarrow τ=\overrightarrow r x \overrightarrow F=rFsinθ$$

Where in this case $$θ$$ is the angle between the force $$F$$ and displacement $$r$$ vectors, and the final term in the equation is the magnitude of the torque.

BUT IT DOESN'T STOP THERE.

If weight = mass * acceleration due to gravity = force (W = mg = F)

And

gravitational potential energy = weight * height = mass * g * height = force * height = MOMENT OF FORCE = WORK DONE BY FORCE. (g.p.e. = mgh = Fh = Fd = M = W )

Concluding that...

Moment of force = work done by force = gravitational potential energy (M = W = g.p.e.)

Wrong conclusion.

Torque (moment) is the cross product of the force and displacement vectors, the third equation above. Since the angle between the force of gravity and the displacement $$h$$ is zero, there is no torque or moment involved and thus no work done by a torque or moment of force when gravity does work on a falling object.

Hope this helps.