ABSTRACT
SOLUTION: If the center of mass of an object lies on the axis of rotation, that object is in balance around that axis, and it will never require any lifting torque. In this setup, the centers of mass of the turntable and the wall can be assumed to be on the axis of rotation, and for the purposes of lifting torque calculations, they can be ignored. The location of the center of mass of all the other pieces combined together is not obvious, and so that location needs to be calculated. With the turntable and wall removed, a Cartesian x and y coordinate system is drawn onto the furniture plan. Its origin, the (0,0) point, must be at the turntable’s axis of rotation. The alignment of the x and y coordinate axes to the collection of furniture shown here is totally arbitrary, any other choice will result in the same end result. Each piece of furniture has a given weight, and a set of x and y distances from the axis of rotation to each object’s center of mass. Because the weight of the kitchen chairs is probably minor relative to the weight of the other pieces, and to simplify seven pieces down to five, the weight of those chairs has just been added into the weight of the table. Calculation of the and
distances for the collection of furniture can now be done. The for-
x1 x2 ..., , y1 y2 ..., ,
m1 m2 ..., ,
mtotal
xc of m yc of m
mulas here have been written in weight units instead of mass since the weight of the pieces is given. In the x direction:
In the y direction:
Finally these x and y coordinates can be converted to a single radial distance between axis of rotation and center of mass by using the Pythagorean theorem
This result puts the center of mass roughly in the center of the back of the couch, and it is not a large distance, meaning the turntable, even with the furniture in place is not far out of balance. The role this dimension has is as the radius in the lifting torque formula, and that will be fully described just below.
