# What is torque, really? and can it be determined for a point on a circle that is away from the center of the rotation axis by a radius $r$?

Let's picture a Rolling without slipping Wheel, that constantly accelerates. radius of a wheel/circle is $$r$$.

Basically any wheel of a vehicle is rotating around its fixed axis - AoR -(axis of rotation)

Most of the times wheels have this AoR perfectly in the center of the circle/wheel/cylinder. What I want to understand is how can we describe the torque at the point $$C$$?

Because as I understand we could go about finding torque for the center of the circle - point $$O$$.

I am not sure if it's possible to find the torque for the point $$C$$, and the point $$P$$ that is point on a line tangent to the circle on the surface level.

• Why do you think you cannot find torques about point $P$? Commented Nov 12, 2019 at 4:46
• The wheel is rolling on a horizontal surface, according to your drawing. So what makes it accelerate? Is there an engine connected to the axis, that forces it to turn faster and faster through a direct torque ? Or is the axis connected to some larger object that is accelerated horizontally by some means placed away from the axis ? Commented Nov 12, 2019 at 5:04
• For me it’s not clear which axes is the wheel rotate? and what is the point AoR?, do you have 2 axes of rotation?
– Eli
Commented Nov 12, 2019 at 7:51
• @Alfred Yes the engine connected two both wheels (not seen in the picture) rotates around the same axis as the wheel does. Commented Nov 12, 2019 at 14:06
• @Eli No there's just one axis of rotation that goes through the point $O$, which is the centre of the wheel- motion is created by an engine that rotates around the same axis as the wheel does. Commented Nov 12, 2019 at 14:08

Conceptually, torque is the turning effect of a force. It is dependent upon two factors- one is the size and direction of the force, and the other is the distance between its point of application and the point around which it is causing the rotation.

In the example you give, the rotation of the wheel is actually a series of instantaneous rotations about the point of contact with the ground, so yes you can calculate its acceleration in terms of a torque around the point P. However, aside from gravity and friction, neither of which will cause a wheel to accelerate on a flat surface, you don't show any motive force in your diagram, so it is hard for me to comment any further.

• What is motive force? Commented Nov 12, 2019 at 8:39
• It is whatever force is causing the acceleration. Commented Nov 12, 2019 at 9:18
• @MarcoOcram I updated the problem with a new image: i.imgur.com/satsg1w.jpg Commented Nov 12, 2019 at 15:30
• @MarcoOcram So right now you can see a segway-like Inverted pendulum vehicle moving across the horizontal plane with acceleration a. It has a rod with an engine giving its motive force to the wheels, that in contact with a plane give this vehicle a nice speed, acceleration is still constant. Commented Nov 12, 2019 at 15:45

Torque is best defined as the work per unit angle (as in Joules per radian) that can be done by a force which is acting in a manner which might cause a rotation. As we learned in statics, it can be calculated relative to any point in your system by using the expression r x F.

The torque about P is $$\ T= F*R=m*a*R$$ as per the definition of torque.

• is it wrong? Because Someone downvoted it Commented Nov 12, 2019 at 19:33
• no it is correct, some people just love to downvote. It gives them a good feeling. Commented Nov 12, 2019 at 19:54
• What is $R$ , then? Commented Nov 12, 2019 at 19:59
• The wheel radius. You used r for it. Commented Nov 12, 2019 at 20:38