Question

The moment of inertia of a body rotating about a given axis is $12kg m^2$ in the SI system. What is the value of the moment of inertia in a system of units in which the unit of length is $5cm$ and the unit of mass is $10g$?
A) $2.4 \times 10^3$
B) $6.0 \times 10^3$
C) $5.4 \times 10^3$
D) $4.8 \times 10^5$

Answer 1

Hint: The moment of inertia of an object is a numerical value that can be calculated for any rigid body that is undergoing a physical rotation around a fixed axis. It is based not only on the physical size of the object and the distribution of its mass, but also on the specific configuration of how the object is moving.

Complete step by step solution:
\(Given,I = 12kg{m^2}\\{\rm{\text{For new system}}}\\ \Rightarrow {\rm{unit length = 5cm}}\\{\rm{unit mass = 10gm}}\\ \Rightarrow {\rm{I = 12}} \times {\rm{1000}} \times {\rm{1}}{{\rm{0}}^4}gc{m^2}\\ \Rightarrow I = 12 \times \dfrac{{1000 \times {{10}^3}}}{{25 \times 10}} \times 10g \times 5c{m^2}\\ \Rightarrow I = 48000 \times 10g \times 5c{m^2}\\\therefore I = 4.8 \times {10^5}\)
Additional Information:
Any moving object has kinetic energy. We know how this is translated for a body undergoing conversion, but how about a rigid body undergoing rotation? This may sound complicated because each point of a rigid body has a different velocity. However, we can use angular velocity - which is the same for the entire rigid body - to express the kinetic energy for a rotating object.
Rotational inertia plays a role similar to mass in linear mechanics in rotational mechanics. Actually, the rotational inertia of an object depends on its mass. It also depends on the distribution of that mass relative to the axis of rotation.
It becomes more difficult to change the rotational velocity of a system when a mass moves past the axis of rotation. Intuitively, this is because the mass is now moving around the circle with greater speed (due to higher speed) and because the momentum vector is changing more rapidly. Both these effects depend on the distance from the axis.
Rotational inertia is given the symbol III. For a single body such as the tennis ball of mass m, rotating at radius r from the axis of rotation the rotational inertia is $I$ = $mr^2$.

Note: The parallel axis theorem allows us to find the moment of inertia of an object about a point o as long as we know the moment of inertia of the shape around its centroid c, mass m and distance d between points o and c.
$Io=Ic + md^2$
Moment of Inertia is the inherent property of matter in rotating motion, the moment of inertia of a body is a measure of its inertia. The greater the moment of inertia, the greater the torque required to create the given angular acceleration.