Understanding Torque and Moment of Force: Explained with Energy Interpretation

[zezima1]

The way that I can understand how torque works is to say that the greater distance from the rotation axis that you apply your force, the greater power you are delivering to your object - since the farther you are away from the rotational centre the faster something is moving on your rotating object.
But the way my book derives the relationship
(1) Ʃτ = Iα
is kind of mysterious to me, as I don't see where my interpretation shows up during the steps.
They say let F be a force applied to the object. Every i'th particle will experience an acceleration such, that the angular velocity is the same for our rigid body. We thus have:
Fi = mi * a = mi * r * α
And multiplying with r yields:
τi = mi * r^2 * α
And summing over all torques gives us (1) - i.e. the rotational analogue of Newtons 2nd law. All there is left to show is then that the internal torques add to zero, but that's simple. I just don't see where the energy interpretation appears in this derivation - can someone open my eyes? :)

 

[eljose79]

Hello, thank you for your question. I can understand how this derivation may seem mysterious to you, but let me try to break it down for you.

First, let's define some terms. Torque is the measure of a force's tendency to cause rotation, and it is calculated by multiplying the force by the distance from the point of rotation. In other words, the farther away the force is applied from the point of rotation, the greater the torque.

Now, let's look at the equation Ʃτ = Iα. This is the rotational equivalent of Newton's second law, which states that the sum of all torques acting on an object equals the object's moment of inertia (I) multiplied by its angular acceleration (α). In other words, the total torque acting on an object is equal to the object's resistance to rotation (moment of inertia) multiplied by how quickly it is rotating (angular acceleration).

In your interpretation, you mentioned that the greater the distance from the rotation axis, the faster something is moving on the rotating object. This is correct, as the farther away from the rotation axis a force is applied, the greater the linear velocity of the object. However, in rotational motion, we are interested in the object's angular velocity, which is determined by its moment of inertia. This is where the energy interpretation comes in.

Let's take a closer look at the equation τi = mi * r^2 * α. This is the torque acting on the i'th particle, which is equal to the mass of the particle (mi) multiplied by the distance from the rotation axis (r) squared, and the angular acceleration (α). Now, if we multiply this equation by r, we get τi * r = mi * r^3 * α. This is the same as saying τi * r = mi * (r^2 * α) * r. Notice that r^2 * α is the angular momentum of the particle, and r is the distance from the rotation axis. Therefore, τi * r represents the work done by the force on the i'th particle, which is equal to the change in the particle's angular momentum.

Now, if we sum over all the particles in the object, we get Ʃτ * r = I * α. This is the total work done by all the forces on the object, which is equal to the change in the object's total angular momentum. This is where