0.32 Phy1320: angular momentum -- the mathematics of torque  (Page 4/13)

Rotational inertia or moment of inertia

Define the rotational inertia, or the moment of inertia, whichever term you prefer as

I = sum from i=0 to i=N(mi*ri^2)

where

• I represents the rotational inertia or the moment of inertia
• mi represents the mass of the ith point particle
• ri represents the distance of the ith point particle from the axis of rotation

Rotational kinetic energy based on rotational inertia

From this, we can determine that

Ks = (1/2)*I*w^2

where

• Ks represents the kinetic energy for the system
• I represents the rotational inertia for the system
• w represents the angular velocity of the system

Similar to translational kinetic energy

Note the similarity between the kinetic energy of a rotating system and the kinetic energy anobject undergoing translational motion:

Ke = (1/2)*I*w^2

Kt = (1/2)*m*v^2

where

• Kr represents the kinetic energy for the rotating system
• I represents the rotational inertia for the rotating system
• w represents the angular velocity of the rotating system
• Kt represents translational kinetic energy
• m represents the mass of an object undergoing translational motion
• v represents the velocity speed of the object undergoing translational motion

Similarities between rotational inertia and mass

The mass of an object tells us how its kinetic energy is related to the square of its velocity.

The rotational inertia of a rotating object tells us how its kinetic energy is related to the square of its angular velocity.

The rotational inertia plays the same role in rotational motion that mass plays in translational motion.

Differences between mass and rotational inertia

While mass and rotational inertia play similar roles, there are also major differences between the two including:

• Rotational inertial depends not just on the total mass of an object, but also on the geometric distribution of the mass within the object.
• Rotational inertia also depends on the choice of the rotation axis, because distances are measured relative to that axis.

Torque from a mathematical viewpoint

Consider a point mass that is constrained to move in a circle. Let the mass be acted upon by an arbitrary force F. We learned earlier that in order for themass to be moving in a circle, there must be a component of force, (the centripetal force), that is directed toward the center of the circle.

Tangential force, mass, radius, and angular acceleration

If we assume that the speed of the mass is changing, there must also be a component of the force that is tangential to the circle at the location of thepoint mass acting on the mass. This force is required to produce acceleration. Therefore, we can write:

Ft = m*y

From above ,

y = r*a

Substitution yields

Ft = m*r*a

Tangential force, radius, angular acceleration, and moment of inertia