Read this to understand angular momentum, how torque plays a role, and how angular momentum is conserved without torque.
No information is given in the statement of the problem; so we must look up pertinent data before we can calculate . First, according to Figure 10.12, the formula for the moment of inertia of a sphere is
so that
Earth's mass is and its radius is . The Earth's angular velocity ω is, of course, exactly one revolution per day, but we must covert to radians per second to do the calculation in SI units.
Substituting known information into the expression for and converting to radians per second gives
Substituting rad for rev and for 1 day gives
This number is large, demonstrating that Earth, as expected, has a tremendous angular momentum. The answer is approximate, because we have assumed a constant density for Earth in order to estimate its moment of inertia.
When you push a merry-go-round, spin a bike wheel, or open a door, you exert a torque. If the torque you exert is greater than opposing torques, then the rotation accelerates, and angular momentum increases. The greater the net torque, the more rapid the increase in . The relationship between torque and angular momentum is
This expression is exactly analogous to the relationship between force and linear momentum, . The equation is very fundamental and broadly applicable. It is, in fact, the rotational form of Newton's second law.