Learning Goal: To understand the definition and the meaning of moment of inertia

Question

Learning Goal:

To understand the definition and the meaning of moment of inertia; to be able to calculate the moments of inertia for a group of particles; to relate moment of inertia to kinetic energy.

By now, you may be familiar with a set of equations describing rotational kinematics. One thing that you may have noticed was the similarity betweentranslational and rotational formulas. Such similarity also exists in dynamics and in the work-energy domain.

For a particle of mass m moving at a constant speedv, the kinetic energy is given by the formula K=12mv2. If we consider instead a rigid objectof mass m rotating at a constant angular speed , the kinetic energy of such an object cannot be found by using the formula K=12mv2 directly, since different parts of the object have different linear speeds. However, they all have the same angularspeed. It would be desirable to obtain a formula for kinetic energy of rotational motion that is similar to the one for translational motion; such a formula would include the term 2 instead of v2.

Such a formula can, indeed, be written: For rotational motion of a system of small particles or for a rigid object with continuous mass distribution, the kinetic energy can be written as

K=12I2.

Here, I is called the moment of inertia of the object (or of the system of particles). It is the quantity representing the inertia with respect to rotational motion.

It can be shown that for a discrete system of nparticles, the moment of inertia (also known asrotational inertia) is given by

I=ni=1mir2i.

In this formula, mi is the mass of the ith particle and ri is the distance of that particle from the axis of rotation.

Part E

Find the total moment of inertia Iy of the system of two particles shown in the diagram with respect to the y axis.

Express your answer in terms of m and r.

Thanks!

Explanation / Answer

total moment of inertia is I

I = m r2

= a ( 3 r )2 + b r2

but considering the mass of 'a' is m

and

the mass of 'b' is 2m and

then rewriting the value

in the above equation

= m ( 3 r )2 + 2 m r2

= 9 m r2 + 2 m r2

= 11 m r2

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