Two masses are balanced on the scale pictured above. If the bar connecting the two masses is
horizontal and massless, what is the weight of mass m in terms of M?Since the scale is not rotating, it is in equilibrium, and the net torque acting upon it must be zero.
In other words, the torque exerted by mass M must be equal and opposite to the torque exerted by
mass m. Mathematically,
Because m is three times as far from the axis of rotation as M, it applies three times as much
torque per mass. If the two masses are to balance one another out, then M must be three times as
heavy as m.
Newton’s Second Law
We have seen that acceleration has a rotational equivalent in angular acceleration, , and that
force has a rotational equivalent in torque,. Just as the familiar version of Newton’s Second Law
tells us that the acceleration of a body is proportional to the force applied to it, the rotational
version of Newton’s Second Law tells us that the angular acceleration of a body is proportional to
the torque applied to it.
Of course, force is also proportional to mass, and there is also a rotational equivalent for mass: the
moment of inertia, I, which represents an object’s resistance to being rotated. Using the three
variables, , I, and , we can arrive at a rotational equivalent for Newton’s Second Law:
As you might have guessed, the real challenge involved in the rotational version of Newton’s
Second Law is sorting out the correct value for the moment of inertia.
Moment of Inertia
What might make a body more difficult to rotate? First of all, it will be difficult to set in a spin if it
has a great mass: spinning a coin is a lot easier than spinning a lead block. Second, experience
shows that the distribution of a body’s mass has a great effect on its potential for rotation. In
general, a body will rotate more easily if its mass is concentrated near the axis of rotation, but the
calculations that go into determining the precise moment of inertia for different bodies is quite
complex.
MOMENT OF INERTIA FOR A SINGLE PARTICLE
Consider a particle of mass m that is tethered by a massless string of length r to point O, as
pictured below: