145 votes
Accepted

Toilet paper dilemma

I'll propose a theory, and I'll describe an experiment I did to test it. Both suggest that the "over" configuration is better, at least if the goal is to make the squares easier to rip off ...
Chiral Anomaly's user avatar
44 votes
Accepted

Why can we calculate moment of inertia, but not inertia?

Classically, the inertia of something is just its mass. If you want an analogous equation, just integrate the mass density $\rho$ of the object over the volume of the object: $$m=\iiint \text dm=\...
BioPhysicist's user avatar
  • 56.4k
41 votes
Accepted

Do black holes have a moment of inertia?

The angular velocity of a Kerr black hole with mass $M$ and angular momentum $J$ is $$ \Omega = \frac{J/M}{2M^2 + 2M \sqrt{M^2 - J^2/M^2}} $$ The moment of inertia of an object can be thought of as ...
gj255's user avatar
  • 6,375
41 votes
Accepted

Why certain rotations are unstable? (Euler Equations)

There is another nice way of seeing this mathematically. It is not too hard to show that in the body frame, there are two conserved quantities: the square of the angular momentum vector $$ L^2 = L_1^...
Michael Seifert's user avatar
34 votes

How many types of inertia are there?

My suggestion is you throw this book away, and use only "good books". Whoever wrote this book has a very confused mind. I can vaguely guess what the author might have had in mind when he ...
Alfred's user avatar
  • 4,204
30 votes
Accepted

Does a rotating rod have both translational and rotational kinetic energy?

It depends what you consider to be the "pivot" about which the rotational kinetic energy is calculated here: If you choose the pivot as the end of the rod that is physically held in place, ...
Uyttendaele's user avatar
28 votes
Accepted

Can the direction of angular momentum and angular velocity differ?

Consider a thin rectangular block with width $w$, height $h$ resting along the xy plane as shown below. The mass of the block is $m$. The mass moment of inertia (tensor) of the block about point A ...
John Alexiou's user avatar
  • 38.4k
25 votes

Why doesn't this way of calculating the moment of inertia of a sphere work?

The moment of inertia is defined relative to an axis, not a point. Therefore the distance you need is from a mass element in the sphere to the $z$-axis, not to the origin. This is why the disk method ...
Andrew's user avatar
  • 48.7k
24 votes
Accepted

Apparent contradiction between moment of inertia and Archimedes's law of the lever

You are making an interesting subtle version of the confusion between mass and weight. Archimedes's law of the lever is equivalently a statement about torques. It is not about $$(2\text{ kg})(1\text m)...
naturallyInconsistent's user avatar
23 votes
Accepted

How come ballerina rotate so fast when there is no external force?

They initially kick the ground and receive an equal and opposite force from it (Newton III), that's where the initial torque comes from. They would not be able to get this from a frictionless surface. ...
SuperCiocia's user avatar
  • 24.6k
23 votes

How does the parallel axis theorem explain the opening of a door?

How does the parallel axis theorem explain the opening of a door? - It doesn't. The moment of inertia of a door does not depend on where you push it. The torque that you apply to a door depends on the ...
Farcher's user avatar
  • 95.9k
22 votes

Does a rotating rod have both translational and rotational kinetic energy?

You have to understand that concepts like rotational kinetic energy are just shortcuts to solve problems efficiently. In Classical Mechanics, we start by defining concepts like kinetic energy on point ...
Noumeno's user avatar
  • 4,474
22 votes

Why can we calculate moment of inertia, but not inertia?

BioPhysicist is right. Your confusion comes from the fact that the word "inertia" is not a "technical" term. It refers to a notion that can apply to many things, physical or ...
Alfred's user avatar
  • 4,204
18 votes

Why do non-rigid bodies try to increase their moment of inertia?

This happens to an isolated rotating system that is not a rigid body so energy dissipation happens inside. Inside such a body (for example, a steel chain in free fall in vacuum) the parts move ...
Ján Lalinský's user avatar
17 votes
Accepted

Does the term "diatomic ideal gas" make any sense?

Of course it does. It helps a little bit to compare the ideal gas to a model that does take note of the size of the molecules and the forces the exert on one another. The van der Waals gas has ...
dmckee --- ex-moderator kitten's user avatar
17 votes

Does a rotating rod have both translational and rotational kinetic energy?

Since the rod's center of mass is changing, does this mean that it also has translational kinetic energy? Yes. You have both translational kinetic energy and rotational kinetic energy. The ...
Bob D's user avatar
  • 71.7k
16 votes
Accepted

Moment of Inertia of an Equilateral Triangular Plate

I can confirm your result. I can also suggest you a neater way to derive it inspired by David Morin - Introduction to Classical Mechanics, check it out in a library if you have access. The main idea ...
mbejan's user avatar
  • 148
15 votes
Accepted

Why do non-rigid bodies try to increase their moment of inertia?

The moment-of-inertia is found via: $$I=\sum m r^2.$$ The fact that each particle tries to increase its distance $r$ to the rotation centre during rotation is a different topic — that is the ...
Steeven's user avatar
  • 50.9k
15 votes

Why certain rotations are unstable? (Euler Equations)

There's an alternative to @MichaelSeifert method which uses angular momentum and moments of inertia: it is to deal with the vector $\vec\omega$ directly as we are interested in the evolution of this ...
ZeroTheHero's user avatar
  • 45.5k
15 votes
Accepted

Why isn't the moment of inertia of a cylindrical tube the difference of those of two cylinders?

In the formula $I_{\rm solid}= \frac 12 mr^2$ the "$m$" is $m_{\rm solid}=\rho \pi r^2L$, where $L$ is the length and $\rho$ the density. So $$ I_{\rm solid}[r]= \frac 12 \rho \pi L r^4. $$ ...
mike stone's user avatar
  • 53.1k
14 votes
Accepted

How can a flywheel make engine run smoothly?

By adding flywheel makes engine to take more power to spin the flywheel because of its huge mass. Efficiency of the engine drops very low. I can only see more burden than smoothness. The burden is ...
Bob D's user avatar
  • 71.7k
14 votes
Accepted

Moment of inertia of 1/4 of a sphere

Yes. It is always possible to calculate the moment of inertia of any arbitrary distribution of mass by doing a volume integral, summing up the moment of inertia for every infinitesimal mass $dM=\rho\,...
G. Smith's user avatar
  • 51.5k
13 votes

Why doesn't this way of calculating the moment of inertia of a sphere work?

Andrew's answer explains what's wrong with your reasoning: the first $r^2$ should be replaced by just $x^2+y^2$, since the moment is based on the distance to the $xy$-axis, not the distance to the ...
Carmeister's user avatar
13 votes

Why do we talk about inertia tensor?

A rank 2 tensor is something that relates two vectors. In this case, the MMOI tensor relates the rotational velocity vector to the angular momentum vector. Given a solid whose internal particles are ...
John Alexiou's user avatar
  • 38.4k
12 votes
Accepted

Moment of inertia: why $\mathrm{dI}=r^2\mathrm{dm}$ instead of $\mathrm{dI}=m\mathrm{dr^2}$?

If the body were a discrete set of point mass objects rotating around an axis, you would write $I=\sum_j m_j r_j^2$ For a continuous body the sum goes over into an integral. You consider it as ...
RogerJBarlow's user avatar
  • 10.1k
12 votes
Accepted

Physical intuition for the parallel axis theorem

True the distance from the axis has decreased but notice that we are taking the square of the square of the distances and hence the signs go away. That is, even if the average change of a collection ...
tryst with freedom's user avatar
11 votes

Why doesn't this way of calculating the moment of inertia of a sphere work?

That's a good, well-stated question and the premise is indeed correct: your independent approach has failed. It fails because you're using $r$ for the distance of the mass of a spherical shell of ...
David's user avatar
  • 703
10 votes
Accepted

How different can the directions of angular momentum and angular velocity be?

As it happens, there are indeed limits to how different the directions of $\vec L$ and $\vec \omega$ can be. This is because the moment of inertia tensor is something called positive semidefinite, ...
Emilio Pisanty's user avatar
10 votes
Accepted

Understanding the differential in integrals

You're right that there's something subtle going on here. For example, let's take the simpler case of defining the total mass an an integral, $M = \int dm$. This has the simple intuitive meaning of ...
knzhou's user avatar
  • 102k

Only top scored, non community-wiki answers of a minimum length are eligible