# Torque required to rotate arm with mass at end

I have to calculate the required torque to rotate a point mass of 15N on the end of an 1.2m long arm which I have calculated to be 10kg in mass. The rotation must have a constant angular velocity of 50rad/s. I am just wondering how to calculate the required torque at the joint. I am also wondering how to deal with the inertia and acceleration of the point mass and the rotating arm which relates to torque required. Any help with calculations for this problem would be greatly appreciated.

First let's consider the peak torque required, important for sizing the motor.

The peak torque will be max when the arm is horizontal, rising. It will be the sum of torques due to the point mass plus that of the arm itself.

Your point mass applies a torque of 15N * 1.2m = ____ N-m.

If the arm's mass is 10kg, its weight is ____ N.

You can model the arm as a point mass at the center of mass.

If it is uniform, the center of mass is at the center of the arm, 0.6 m.

The arm applies a torque of [arm weight] N * 0.6 m = _____N-m.

As long as the actuator can apply torque greater than that, the machine will (eventually) accelerate to the required angular velocity.

Now let's move on to the instantaneous torque, as a function of time, in order to maintain a constant angular velocity.

Let theta be the angle the arm makes with the up-vertical (clock hands at midnight). The moment arm for each of the above torques is proportional to sin theta. So the instantaneous torque will be a sinusoid. Note that half the time the actuator is lifting the weight and the other half it is holding back the weight to keep it from falling freely. This is consistent with sin theta being negative half the time.