# Deriving Moment of Inetia using just linear dynamics

• NANDHU001
In summary, moment of inertia can be derived using linear dynamics and calculus by considering a mass being accelerated and rotating in a circle. The torque can be calculated by multiplying the force by the radius, and by integrating this over the total mass, the total torque can be obtained. This approach is an alternative to the traditional method of using the energy equation and the analogy of 1/2mr^2w^2 with 1/2mv^2.
NANDHU001
Can moment of inertia be derived using just linear dynamics and calculus. Textbooks usually derive moment of inertia using energy equation and and analogy of 1/2mr^2w^2 with 1/2mv^2. I would like to know if it can be approached in a different manner using just linear dynamics.

Yes, If you consider a mass being accelerated and rotates in a circle.
Then the acceleration is:

$F=ma$
multiply both sides by r:

$\tau=rma=r^{2}m\alpha$
where $\alpha$ is the angular acceleration.
Take this sum of all masses:

$\sum r^{2}dm$

Or another way:

The force on a small element dm is:
$dF=r\frac{d\omega}{dt}dm$
then the torque on this small mass dm is:
$d\tau= rdF=r^{2}\frac{d\omega}{dt}dm$
integrating this over the total mass gives the total torque:
$\tau=\int r^{2}dm\frac{d\omega}{dt}$

Hope it helps

## 1. What is moment of inertia and why is it important in linear dynamics?

Moment of inertia is a physical property of an object that describes its resistance to rotational motion. In linear dynamics, it is important because it relates to an object's mass distribution and affects its rotational acceleration.

## 2. How is moment of inertia determined using just linear dynamics?

Moment of inertia can be determined by using the equation I = mr^2, where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation. This equation is derived from Newton's second law of motion.

## 3. Can moment of inertia be calculated for all objects using linear dynamics?

No, moment of inertia cannot be calculated for all objects using just linear dynamics. This method is most accurate for objects with a simple shape and a single axis of rotation. For more complex objects, other methods such as integration or computer simulations may be necessary.

## 4. How does changing the mass distribution of an object affect its moment of inertia?

Changing the mass distribution of an object can greatly affect its moment of inertia. For example, if the mass is concentrated at the edges of an object, it will have a higher moment of inertia and will be more difficult to rotate compared to an object with mass distributed evenly.

## 5. What are some real-world applications of understanding moment of inertia through linear dynamics?

Understanding moment of inertia is important for many real-world applications, such as designing vehicles and machinery. It is also crucial for predicting the behavior of rotating objects in physics and engineering, such as spinning tops, pendulums, and gyroscopes.

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