Define moment of inertia. Write the parallel and perpendicular axis theorem.

Derive an expression for moment of inertia of a disc of radius r, mass m about an axis along its diameter.

Q: 1. Define moment of inertia. Write the parallel and perpendicular axis theorem. 2. Derive an expression for moment of inertia of a disc of radius r, mass m about an axis along its diameter.

1. Moment of inertia is inertial equivalent in rotational motion. Moment of inertia of a rigid body is defined as the sum of the products of the constituent masses and the squares of the perpendicular distance from the axis of rotation. If $m_1, m_2...m_n,$ are the masses at perpendicular distances $r_1, r_2...,r_n$ then, moment of inertia $\text{I}=\text{m}_{\text{l}}\text{r}^2_{\text{l}}+\text{m}-2\text{r}^2_2+\dots+\text{m}_{\text{n}}\text{r}^2{\text{n}}$ $=\sum\limits_{\text{i}=\text{l}}^\limits{\text{n}}\text{m}_{\text{i}}\text{r}^2_{\text{i}}$ It is measured in $kgm^2$ and has the dimensions of $ML_2$. Parallel Axis theorem: The moment of inertia about passing parallel to the axis through the centre of mass of a rigid body is the sum of the moment of inertia $(I_{cm}.)$ of the body about the axis through centre of mass and the product of its mass M and square of the separation $(a^2)$ between the parallel axis. i.e., $I = I_{cm} + Ma^2$ Perpendicular Axis theorem: If $I_x$ and $I_y$ are the moment of inertia about the x and y-axis of any rigid mass, then the moment of inertia I_z about z-axis is the sum of $I_x$ and $I_y$. $I_z = I_x + I_y$ 2. Moment of intertia of a uniform circular disc about its diameter: Using the theorem of perpendicular axes, we get $I_d + I_d$ = Moment of inertia of the disc about perpendicular axis yoy' $\text{i.e.,}2\text{I}_{\text{d}}=\frac{1}{2}\text{MR}^2\text{ or }\text{I}_{\text{d}}=\frac{1}{4}\text{MR}^2$

Question

Define moment of inertia. Write the parallel and perpendicular axis theorem.
Derive an expression for moment of inertia of a disc of radius r, mass m about an axis along its diameter.

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Solution

Moment of inertia is inertial equivalent in rotational motion. Moment of inertia of a rigid body is defined as the sum of the products of the constituent masses and the squares of the perpendicular distance from the axis of rotation. If $m_1, m_2...m_n,$ are the masses at perpendicular distances $r_1, r_2...,r_n$ then, moment of inertia

$\text{I}=\text{m}_{\text{l}}\text{r}^2_{\text{l}}+\text{m}-2\text{r}^2_2+\dots+\text{m}_{\text{n}}\text{r}^2{\text{n}}$
$=\sum\limits_{\text{i}=\text{l}}^\limits{\text{n}}\text{m}_{\text{i}}\text{r}^2_{\text{i}}$
It is measured in $kgm^2$ and has the dimensions of $ML_2$.
Parallel Axis theorem: The moment of inertia about passing parallel to the axis through the centre of mass of a rigid body is the sum of the moment of inertia $(I_{cm}.)$ of the body about the axis through centre of mass and the product of its mass M and square of the separation $(a^2)$ between the parallel axis. i.e., $I = I_{cm} + Ma^2$

Perpendicular Axis theorem: If $I_x$ and $I_y$ are the moment of inertia about the x and y-axis of any rigid mass, then the moment of inertia I_z about z-axis is the sum of $I_x$ and $I_y$.
$I_z = I_x + I_y$

Moment of intertia of a uniform circular disc about its diameter:

Using the theorem of perpendicular axes, we get $I_d + I_d$ = Moment of inertia of the disc about perpendicular axis yoy'
$\text{i.e.,}2\text{I}_{\text{d}}=\frac{1}{2}\text{MR}^2\text{ or }\text{I}_{\text{d}}=\frac{1}{4}\text{MR}^2$

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