# What is Rotational inertia: Definition and 181 Discussions

The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

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1. ### Rotational inertia of square about axis perpendicular to its plane

For this problem, How do we calculate the moment of inertia of (2) and (3)? For (3) I have tried, ##I_z = \int r^2 \, dm ## ## ds = r ## ##d\theta ## ##\lambda = \frac {dm}{ds}## ##\lambda ## ##ds = dm ## ## \lambda r ## ##d\theta = dm ## ##I_z = \lambda \int r^3 d\theta ## ##I_z = \lambda...
2. ### I Rolling balls with different rotational inertia

I am stuck on problem presented about putting golf balls. A stationary golf ball (mass=45g, dia=42mm, solid & homogeneous) is struck by a horizontal force (putter) and ignoring sliding immediately starts rolling on a level putting green. The ball eventually stops due to rolling resistance...
3. ### What's the source of increase in rotational energy of carousel?

A carousel has the shape of a circular disc with radius 1.80 m and a mass of 300 kg. There are two people with masses of 30 and 45 kg out on the edge while carousel rotates with the angular speed 0.6 rad / s. The people move towards the center of the carousel Calculations show that the...
4. ### B Is the Universe Expanding Due to Spinning?

Hello all, I recently did a thought experiment and thought, "what if the universe as a whole is spinning?" This could solve the dark energy problem, as if the universe was spinning, there would be outward pull, and therefore keep the universe expanding. And, I don't know if this means anything...
5. ### Pendulum, Rotational Inertia and Center of mass

This is the figure given. My Attempt ##T=2\pi \sqrt\frac {I}{0.5gd}## ##\frac {m_r} {l} ## ##dm= \frac {m} {l}dx## ##dI = dm_r x^2## ##dI=(\frac {m_r} {l}dx)x^2## ##I= \int_l^0 (\frac {m_r} {l}dx)x^2 \, dx ## ##I_c.m=\frac {m_r l^2}{3}## ##I_,, = \frac {m_r l^2}{3}+m_p x^2## Given...
6. ### Rotational Inertia: Hoop vs Disk

I know that a hoop should have a higher rotational inertia than a solid disk because its mass is distributed further from the axis of rotation. What I don't understand is how a disk of the same mass and radius can have a higher rotational inertia. If the objects roll freely their axes of...
7. ### Rotational Inertia and Net Torque with Friction

I converted the amount of rotations completed in 5 seconds into radians. 23.4 rot * 2pi = 147 rad I found the angular acceleration of the object in the first 5 seconds it was speeding up. Wf = Wi + at a = 5.881 rad/s^2 I then used the moment of inertia given in the problem to solve for torque. T...
8. ### Calculate the rotational inertia of a solid hexagonal

What I did: ##\frac{dm}{dA} = \frac{M}{\frac{3\sqrt3 R^2}{2}}## ##dm = \frac{2M}{3\sqrt3 R^2} dA## (1) ##dA=3\sqrt3 rdr## (2) (2) in (1) ##dm = \frac{2M}{3\sqrt3 R^2} 3\sqrt3 rdr## Now in the integral ##I = \int \frac{r^2 2Mrdr}{R^2}## How can I solve the integral interval? I think I...
9. ### Dumb question about inertia and rotational inertia

Consider two bodies, A and B, of equal mass set at a short distance. Body A is spinning and body B is at rest. Then, through some kind of electromagnetic device, a strong repulsive force is established between them. Will both be displaced at the same speed?
10. ### Rotational inertia - globes connected by a thin rod

Problem Statement: Finding the rotational inertia Relevant Equations: I=∑m*r^2 A rigid body of 2 massive globes with homogenous mass distribution and a thin rod is connecting the 2 globes. The globes has radius R1 = 0.18 m and R2 =0.28 m and masses m1=193 kg and m2=726 kg. The thin rod has...
11. ### Rotational Inertia: Calc. & Guidance Needed

This question is ridiculously complex for me. I know how to find the Inertia if it was a rod ##I =\frac{1}{12} ML^{2} = \frac{1}{12}(0.000104)(1.6900)^2 =0.00002475 kg * m^2## To find the Inertia for the individual disks I am having trouble. Not sure how to add them all up together. Guidance...
12. ### Calculating rotational inertia

Homework Statement How come I can't get the correct answer using Energy as a way to solve this? Homework Equations 3. The Attempt at a Solution [/B] The answers use conservation of momentum which makes perfect sense and I understand that, however I used an energy approach where E(flywheel) =...
13. ### Rotational inertia hoop around a central axis

Homework Statement Homework Equations I= mr^2 for a hoop around a central axis I= 1/12 (m)(l)^2 for thin rod about axis through center perpendicular to lenght The Attempt at a Solution I am totally confused. i said first that the three masses each will make a hoop shape so i found I=mr^2...
14. ### Linear Momentum to Angular Momentum

Homework Statement A 10 g bullet traveling at 400 m/s strikes a 10 kg , 1.2-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door immediately after impact? Homework Equations p[/B]= mv L = Iω...
15. C

### What is the Speed of a Falling Object Attached to a Rotating Spherical Shell?

Homework Statement A uniform spherical shell rotates about a vertical axis on frictionless bearings. A light cord passes around the equator of the shell, over a light, frictionless pulley, and is attached to a small object that is otherwise free to fall under the influence of gravity. Calculate...
16. ### Angular momentum and rotational inertia

Homework Statement You decide to design a bicycle that will have only 3/4 of the angular momentum of the original wheel when both wheels are traveling along a road at the same velocity. The original wheel had a diameter of d1=37cm and rotational inertia of I1=0.32 m2kg. If your new wheel has a...
17. S

### Rotational Inertia and Angular Momentum

Homework Statement A disk with a rotational inertia of 8.38 kg·m2 rotates like a merry-go-round while undergoing a torque given by τ = (5.03 + 1.01t) N · m. At time t = 1.00 s, its angular momentum is 6.57 kg·m2/s. What is its angular momentum at t = 3.00 s? Homework Equations dL/dt= T...
18. ### Other symbols for rotational inertia?

So I'm currently doing a project on motors. It just so happens that I'm dealing with both electrical current and rotational inertia. I have one small problem. The symbol for electrical current is I. But so is rotational inertia! Are there any other symbols for rotational inertia/electrical...

42. ### Rotational inertia: a contradiction?

We know that the rotational inertia I of a certain object is I =∫r∧2 dm where r is the distance between the axis of rotation and the increment of this object that carries a mass dm. What confuses here is the following: Take for example a hoop of mass M and radius R. Integration theory gives...
43. ### Rotational Inertia of a triangle

Homework Statement A thin, uniform vane of mass M is in the shape of a right triangle, as shown. Find the rotational inertia about a vertical axis through its apex, as shown in the figure. Express your answer in terms of the triangle’s base width b and its mass M. Homework EquationsThe...
44. ### Calculating the Force on a Bicycle Wheel

I have been trying this problem for a while and can't seem to figure it out: A bicycle wheel has a radius R = 32.0 cm and a mass M = 1.82 kg which you may assume to be concentrated on the outside radius. A resistive force f = 137 N (due to the ground) is applied to the rim of the tire. A force...
45. ### Calculating Rotational Inertia: Hoops vs. Solid Cylinders

Homework Statement Homework EquationsThe Attempt at a Solution I assume the energy stored is = 1/2 (I) (ω^2) I (moment of inertia) is MR^2 since it's a hoop? or is it a solid cylinder? do we need to convert the rpm (revolutions per minute) to radians per sec?
46. ### Rotational Inertia of a Pulley System

Homework Statement Homework Equations Kinetic Energy = 0.5 I w^2 v = r * w torque = Inertia * angular acceleration The Attempt at a Solution That's the problem. I have no idea where to start. I assume that the end goal is to find the angular velocity, and convert to linear velocity, but I...
47. ### Rotational inertia of a rod that has mass on one end

Homework Statement a 3 kg mass is on the end of a metal rod which is pivoted at one end. the mass of the rod is 2kg its length is 4 meters Homework Equations I=(ML^2)/3 The Attempt at a Solution the rotational inertia of the rod itself is 10.6 but i don't know how the 3kg mass at the end...
48. ### Solve Rotational Collision Homework: 100 RPM to 50 RPM

Homework Statement A 2kg , 0.2 m diameter turntable rotates at 100 rpm on frictionless bearings. Two 0.5 kg block fall from above, hit the turntable simultaneously at opposite ends of the diameter, and stick. What is the turntable's angular velocity (in rpm) just after? Homework Equations L= I...
49. ### How Does Rotational Inertia Affect the Motion of a Sphere on a Moving Ramp?

1. Homework Statement A solid sphere (mass of m, radius of r and I=2/5 mr2) is rolling without slipping on a rough surface with a speed of v. A ramp (mass of 2m and angle of θ) rests on a smooth surface and is free to slide on the surface. As the ball rolls up the ramp, the ramp begins to move...
50. ### A wheel with rotational inertia I = 1/2MR^2

Homework Statement [/B] A wheel with rotational inertia I = 1/2MR^2 about its horizontal central axle is set spinning with initial angular speed W0. It is then lowered, and at the instant its edge touches the ground the speed of the axle (and CM) is zero. Initially the wheel slips when it...