MHB What is the concept of moment of inertia?

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Moment of inertia, denoted as $I$, represents an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It is defined by the equation $\alpha = \dfrac{\tau_{net}}{I}$, where $\alpha$ is angular acceleration and $\tau_{net}$ is net torque. The discussion clarifies that moment of inertia is not torque itself but rather a measure of how difficult it is to change an object's rotational state. Further questions will explore its relationship with the Center of Mass, Moments of Inertia, and Radius of Gyration. Understanding moment of inertia is crucial for analyzing rotational dynamics in physics.
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In simple terms, what exactly is meant by moment of inertia as taught in Calculus 3?
 
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Simply put, it is the torque needed for a desired angular acceleration about a rotational axis. :D
 
MarkFL said:
Simply put, it is the torque needed for a desired angular acceleration about a rotational axis. :D

I will post one or two questions regarding this topic.
 
Here is a better explanation sent to me via PM (I simply copy-pasted from Wikipedia):

Moment of inertia, $I$, is not a torque, rather it is the resistance an object has to a change in its state of rotational motion, i.e.

$\alpha = \dfrac{\tau_{net}}{I}$

... mass is its counterpart in the translational world.
 
MarkFL said:
Here is a better explanation sent to me via PM (I simply copy-pasted from Wikipedia):

Moment of inertia, $I$, is not a torque, rather it is the resistance an object has to a change in its state of rotational motion, i.e.

$\alpha = \dfrac{\tau_{net}}{I}$

... mass is its counterpart in the translational world.
I will post three questions tomorrow that involve p = another variable in relation to the Center of Mass and Moments of Inertia and Radius of Gyration.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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