# Can you calculate torque without any pulleys involved, and if so, how?

I'm trying to calculate the type of motor I need to turn something of a certain weight, so I feel the thing I'm trying to calculate is torque, but, when I research how to find torque, I find this "To calculate load torque, multiply the force (F) by the distance away from the rotational axis, which is the radius of the pulley (r). If the mass of the load (blue box) is 20 Newtons, and the radius of the pulley is 5 cm away, then the required torque for the application is 20 N x 0.05 m = 1 Nm. " So that leads me to ask, if I have no pulley, and am directly attaching whatever I want to turn to the axis, then how do I calculate torque without some distance from the axis? If I just multiply it by 0, as in 0m from the axis, then the force needed would be 0, which makes no sense as far as I can see. Is torque even what I should be calculating, and if not, how do I find what I want to know? Thank you!

• What about a lever? then the length of the lever is needed. Feb 4, 2022 at 9:31
• Torque = Rotational inertia * Angular acceleration
– towe
Feb 4, 2022 at 10:04
• The rotational inertia depends on the mass distribution of your object (think light, large flywheel of the same mass as a solid little cylinder)
– towe
Feb 4, 2022 at 10:05
• @towe i see! so rotational inertia * angular acceleration still fits the "length * weight = torque" formula given earlier, correct? of not, how come it changes? – Feb 4, 2022 at 10:36
• @SolarMike Ohhhh so if there's no pulley, and you're directly attaching a lever, then the weight of the lever and the length of the lever is needed rather than the distance from the axis and the weight of something on the pulley? – Feb 4, 2022 at 10:36

When we talk about an external force trying to rotate a certain object at a pivot point, the moment arm (distance away from the rotational axis) comes into the picture. And the expression for torque would then be equal to,

$$T = F.r$$

• T= torque
• F = Force acting on the object.
• r = moment arm (distance away from the rotational axis)

But in your case, you are trying to find the torque of the motor that has to rotate a certain object connected directly to its output shaft. This scenario can be pictured as a force trying to push an object So the linear acceleration is the actual input, which in turn will give you the value of the force required to that object with that acceleration. Now coming back to your question. Refer to the image below. Here, the input would be the angular acceleration which can be calculated using this formula,

$$\theta = \omega_i + \frac{1}{2}\alpha t^2$$ if you know the time and angle to be traveled.

With the angular acceleration known to us, we can easily calculate the torque required using the formula,

$$T = I\alpha$$

$$I$$ being the mass moment of inertia of the object connected to the motor shaft.

$$\alpha=\frac{T}{I}$$
• $$\alpha$$ = angular acceleration