How to calculate gear ratio needed to move a pivoting arm?

Question

There's an arm of length l and mass m, and a motor-driven shaft with a constant torque t.

The arm swivels on a fixed pivot point.

The strategy on the left uses a cable anchored on the far end and the cable is spooled by turning the shaft clockwise to move the arm to a certain angle. The movement happens linearly, deg/s degrees per second.

The setup on the right uses a gear welded onto the pivoting end of the arm. Keeping every other trait of the system the same, how can I resolve the necessary gear ratio to move the arm to the same angle at the same deg/s?

(Motors in both systems should draw equal power from the wall)

Answer 1

The image on the left will decelerate as the arm rotates to vertical because of the change of angle of pull and the distance between the centres of rotation.

To avoid a load of maths I would use CAD.

• Construct an arc from the arm's centre with radius of the arm's length. This shows the path of the tip of the arm.
• Calculate the circumference of the cable spool.
• Construct an arc from the centre of the spool with a radius = length of string - calculated circumference.
• Construct a line between the arm pivot and the intersection of the two arcs.
• Measure the angle between the arm's initial and final positions. Let's call it θ:360. You will, of course have to choose an integer number of teeth.