The nature of torque as a vector

Question

I can’t comprehend torque’s role properly:

1. What is torque’s nature? Is it some kind of energy? Because its unit is ($\mathrm{N\cdot m}$) and ($\mathrm{N\cdot m}$) is similar to (Joules) and it's energy unit.

2. As we all know “Torque” is a vector, so how can we comprehend the role of torque’s direction in the nature? I mean what’s the relation between torque’s direction and natural phenomenons involving with torque? Because as we know torque is the cross product of force and length, however why its direction is not aligned by the direction of the object’s spin? For example as in the uploaded picture, the object falls in the direction of the force while torque's direction is toward the viewer. How can we understand this in the nature and not just mathematical formulas?
   

Answer 1

Torque is a generalized force. Like force, it causes a change in motion (but instead of momentum change, 'angular momentum' change).

we know torque is the cross product of force and length, however why its direction is not aligned by the direction of the object’s spin?

The 'direction of the object's spin' is a preexisting condition, just as velocity, V, is the preexisting condition for Newton's $$ \vec F = m \vec A = m \frac{d{\vec V}}{dt} $$ and torque, being a generalized force, causes alterations of the angular momentum direction (as well as amplitude) of a solid body. Torque is directed according to a cause which is not the spin of the body acted upon.

Gyroscope motion would be a typical example of the utility of the vector picture, and is most enlightening (and useful) when the applied torque is not on the spin axis.

The vector cross product, by convention, obeys a right-hand-rule, so it is clear that torque, like angular momentum, is NOT energy. Torque, even as a scalar (a magnitude, not a vector) always appears in equations with other terms that are likewise determined by a cross product and matching right-hand convention.

Answer 2

Torque is properly a pseudo-vector. Its definition is associated with the concept of Force: in fact torque is defined as: $\vec M= \vec r \times \vec F$, with $\vec r$ and $\vec F$ are vectors with represent, respectively, the position and the force applied... The vector product between two vectors generates another vector: its magnitude is $|\vec M|=|\vec r| \cdot |\vec F| \cdot \sin(θ)$ where $θ$ is the angle between $\vec r$ and $\vec F$; its direction is perpendicular to the plane formed by $\vec r$ and $\vec F$, and its verse is in according to the right hand rule. Now, why introducing this vector? Well, the explanation is really simple... Suppose that you have to open a door and do this experiment: use the same force $\vec F$ and initially apply this force on the door jamb; the second time apply it to the other side... You will observe that although the force is the same, in the second case opening the door is simpler. The reason is that the torque is greater in the second case due to the fact that $\vec r$ (force's distance from fulcrum) is bigger.

And now the answer to your first question... Yeah the torque has the same units Nm of energy... But torque isn't an energy! It is not a number but a vector... You can see it as the equivalent to the force used for rotation.

Bye and good study of physics!

Answer 3

We all know torque is a vector

It isn't a vector. It's usually modelled as such in 3d because there is a unique vector that is orthogonal to the plane of rotation - the axis.

However, in 4d we see that there is no such unique vector. Here we model it using an bivector or antisymmetric 2-tensor that directly models the plane of rotation with it's magnitude corresponding to the size of the torque. This is just as true in 5d or higher.

This is one way of using higher dimensions to clarify concepts in mechanics.

Answer 4

In as rotating object, each point is moving in a different direction. Because of this, the vectors describing rotation are defined as being along the axis of rotation (a semi-fixed direction). This is done in a self consistent manner that requires vector products to relate them to the corresponding linear quantities. As suggested by Mr. Fritch, torque can be defined as the work done per unit angle of rotation, Thus giving unit of Joules/radian.

Answer 5

What is torque’s nature? Is it some kind of energy? Because its unit is (n.m) and (n.m) is similar to (joules) and it’s energy unit.

Rotational quantities (angular displacement, angular velocity, angular acceleration etc.) differ from their corresponding linear quantities in that angles are dimensionless, i.e., the angular unit radian is a dimensionless unit.

So, for example, the unit of angular acceleration is $\mathrm{\frac{rad}{sec^2}}$ with dimensions $\mathrm{T}^{-2}$

Now, torque is the rotational analogue of force and, as such, we expect that rotational work is torque through an angle just as linear work is force through a distance.

However, since angular displacement is dimensionless, torque must have the dimensions of work: $\mathrm{ML^2T^{-2}}$.

But if we think of force as work per distance, torque can be thought as work per angle and so the unit of torque can be thought of as $\mathrm{\frac{J}{rad}}$ to conceptually keep it distinct from the unit of work $\mathrm{J}$. From the BIPM web site:

For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit newton metre, or it may be thought of as energy per angle, suggesting the unit joule per radian

Answer 6

Maybe my answer is not enough of satisfaction as the above 3 but I am going to try my level best to explain it to you.

1) That's actually a really good question. But, no: torque is not some kind if energy. See if you have learnt about impulse and momentum you'll notice that they have the same units and dimensions too right? But they have a difference, so not all things having same units are same. Also torque and energy have the same dimensions and not units!

Their unit is same due to the fact that: $W=\int \vec τ \cdot d\vecθ$

$W$= work done , basically energy

$\vec τ$=torque

$ \vecθ$= angular displacement

when torque is constant we can just write

$W=\vec τ \cdot \vecθ$

we can see that since $\vecθ$ is dimensionless $\implies$ Energy and torque have the same unit

2) The direction of torque can be calculated by right hand rule. It just gives us a "Sense of Motion". It helps us to judge what will be the motion under different torque (you just apply the vector algebra addition). Its direction cannot be aligned by the direction of object's spin as there is none, at every different point you have different direction and magnitude (if distance from origin is different). So we can do that, but it would be a lot of work. Instead take your right hand and point your thumb at the torque and curl your hand, you'll get the direction of motion. Trust me, this is really easy compared to the other way.

Answer 7

As I mentioned in this answer, the nature of torque is that of a force at a distance. Specifically, it tells us where in space the line of action of a force is.

Also, on why isn't torque measured in Joules, since unlike energy which is a force along a displacement, torque is a force at an offset distance.

Answer 8

1. Direction of torque: Axis around which the rotation takes place.

2. Module of torque: Module of contribution to the rotation.