Question

Find out the moment of inertia of the following structure (written as $PHYSICS$ ) about the axis $AB$ made of thin uniform rods of mass per unit length $\lambda $ .
     

A. \[13\lambda {l^3}\]
B. $10\lambda {l^3}$
C. $7\lambda {l^3}$
D. $11\lambda {l^3}$

Answer 1

Hint: In this question, use the moment of inertia concept that is product of mass and square of its perpendicular distance from the origin to calculate the moment of inertia of a latter. The total number of letters in “PHYSICS” is $7$. And multiply the moment of inertia by seven to calculate the total moment of inertia.

Complete step by step answer:
In this problem, first we must know about the moment of inertia. As per definition we know that, moment of inertia is a quantity representing the body’s tendency to resist the angular acceleration, which is the sum of the products of the mass of each particle in the body with square of its distance from the axis of rotation moment of inertia is used to calculate the angular momentum and allows us to explain how is the rotation motion change when the distribution of mass change.

As we know that the mathematical expression of the moment of inertia is,
$I = m{r^2}$

Where, $r$ is the perpendicular distance from the origin and $m$ is the mass.

Also, in this question given that mass per unit length is $\lambda $ and the perpendicular distance from the origin is $l$
So, the mass will be,
$m = \lambda \times l$
It is given in the question that each letter is made up of same rod, so the mass of each latter will be $m = \lambda \times l$

Now we will calculate the moment of inertia of letter $P$ by substituting the values in the moment of inertia equation as,
$I = \lambda l \times {l^2}$
$ \Rightarrow I = \lambda {l^3}$

As we know that the total latter in the word PHYSICS is $7$.
Therefore, the total moment of inertia of the word Physics will be $7\lambda {l^3}$

Therefore, option C is correct.

Note:
The given question is based on the length of the rod as the density of the rod is depend on the length, so we have taken the distance based on the center of the rod length, but if the density depend on the volume of the rod then the distance will be taken on the basis on the center of mass.