Summary for Chapter 4

For objects rotating about a fixed axis, the angular velocity is the rate at which it is rotating, i.e. the magnitude of the angular velocity is the angular speed and that is the number of complete revolutions (rev) is makes per second. If it make 7 revolutions in 2s, the angular speed is 7rev/2s = 3.5 rev/s. (Often motors give the angular speed in revolutions per minute, rpm, instead of per second.)

A rigid object rotating about and axis with no external forces acting on it will continue to rotate, i.e. its rotational motion will not change. What causes the rotation motion to change? The answer is a force, but it is more complicated than that. It depends on WHERE the force is applied. It is a torque that causes the rotational motion of an object to change. If the net torque on an object is zero, its rotational motion will not change. If it is non-zero the rotational motion will change. Therefore for an object to be in equilibrium for rotational motion, the total torque on it must be zero. (Again, you can have dynamic equilibrium [constant rotational motion], or static equilibrium [not moving].)

Torque

For rotations about an axis, the important quantity is called the torque. The magnitude of the torque is the force times the distance from the axis, if F is ^ to d. The torque will tend to rotate the board either clockwise (CW) or counterclockwise (CCW) about the axis. For the example at the right it will rotate it clockwise. I usually call clockwise torques negative and CCW torques positive, so I would say that the torque in the picture is -Fd. If F = 100n and d=0.5m, it would be - 50Nm. To be in balance (not moving) the net torque on a system must be zero.

For the above to be in balance, it would need a CCW torque on it equal in magnitude to Fxd (50Nm) as shown at the right. In this case the two torques cancel, or balance each other and F₁d₁ = F₂d₂. Note that this allows the 100N force to support the 500N force, so I can lift a 500N weight with a 100N force.

Center of Gravity or Center of Mass

For purposes of calculating torques, the gravitation force on an object acts as though all the mass of the object were located at the one point, called the center of gravity or center of mass. The center of mass is usually at or near the geometric center of an object. For a uniform sphere, the center of mass is at the center of the sphere. For a meter stick, it will be at the center of the stick, if the stick is uniform.

If an object is being supported at two points, e.g. a person standing on both feet, the object can only be in equilibrium if the center of gravity is above the area outlined by the two points of support, e.g. the area between the persons feet. It it is not, the torque due to gravity will tip the object, assuming it is not bolted or glued down.

In the fig. A at the right the mustard colored object is resting on the ground. Fig. B shows the region above the base of support as a gray region. If the center of gravity is in that region, the object is stable. If it is outside that region, it is not stable. As shown in fig. B, the center of gravity is above the support base, barely, so it is stable.