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.
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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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) =...
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...
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...
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...
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...
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...
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...
Homework Statement
Side by side on the top of an incline plan with height=2 meters a cylinder (Ic= MR^2/2) and a sphere (Ie=2MR^2/5) with equal radius, that come down to the base, rolling without slipping. Mass of the cylinder = 2.0 kg; Mass of a sphere=4.0 kg.
Homework Equations
$$K_r= 1/2...
Homework Statement
Wheels A and B as shown in the figure are connected by a massless belt that does not slip. The radius of A is R and the radius of B is r. What is the ratio of rotational inertias ##\frac{I_a}{I_b}## if the two wheels had the same angular momentum about their central axes...
A 24 kg solid door is 220 cm tall, 95 cm wide. What is the door's moment of inertia for rotation about a vertical axis inside the door, 17 cm from one edge?
I've looked at examples of how to do this problem. They connect it to center mass but I get confused when determining what the center mass...
Hi,
I wonder if a gravitationally bound system ...like a binary star system (or a spinning galaxy for that matter) has a similar inertia in its rotational axis like your classic spinning top? Is there resistance to a change in the axis?
I appreciate the system is not solid and so not obviously...
Homework Statement
A solid uniform sphere of radius R and mass M has a rotational inertia about a diameter that is given by (2/5)MR^2. A light string of length 3R is attached to the surface and used to suspend the sphere from the ceiling. What is its rotational inertia about the point of...
Homework Statement
Consider one rod of length L which is spun at its center perpendicular to its length; it will have a certain moment of inertia. Now, if the same rod is folded in the middle creating a certain angle and still spun at its center, what happens to its moment of inertia?
Homework...
Homework Statement
The shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L = 1.0000 m and (total) mass M = 100.0 mg. The disks are uniform, and the disk arrangement can rotate about a perpendicular axis through its central disk at point O...
Homework Statement
Prove the formula for inertia of a ring (2D circle) about its central axis.
Homework Equations
I = MR^2
Where:
M: total mass of the ring
R: radius of the ring
The Attempt at a Solution
- So I need to prove the formula above.
- First, I divide the ring into 4...
Homework Statement
If we multiply all the design dimensions of an object by a scaling factor (f), its volume and mass will be multiplied by f^3. a) By what factor will its moment of inertia be multiplied? b) IF a 1/48 scale model has rotational kinetic energy of 2.5 Joules, what will be the...
I've gathered data of a skateboard going up an include and rolling back with a force plate.
- Vertical force
- Velocity
The vertical force graph looks like this:
The first bump is when the skateboard rider hits the incline.
I'm doing an investigation and I don't know how this force and...
Homework Statement
One end of a light spring with force constant k = 100 N/m is attached to a vertical wall. A light string is tied to the other end of the horizontal spring. the string changes from horizontal to vertical as it passes over a pulley of mass M in the shape of a solid disk of...
Homework Statement
Is rotational inertia an intrinsic property of an object?[/B]Homework EquationsThe Attempt at a Solution
So I know that rotational inertia is a property of an object that deals with a resistance to a change in the state of rotational motion but is it an intrinsic property? I...
Homework Statement
This is a question I have for a presentation on rotational motion: "A solid cylinder rolls down an incline faster than a hoop [or say an open cylinder], whether or not they have the same mass or diameter. The hoop has greater rotational inertia relative to its mass than a...
Homework Statement
A rigid body is made of three identical thin rods, each with length L = 0.530 m, fastened together in the form of a letter H, as suggested by the figure here. The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body...
Homework Statement
In the figure, a wheel of radius 0.42 m is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a 2.7 kg box that slides on a frictionless surface inclined at angle θ = 28 owith the horizontal. The box accelerates down the...
I am conducting an experiment that rotates a sealed donut cylinder to investigate amongst other things the inertia of the contained fluid.
Process
I rotate the cylinder to a set speed (assuming that the viscosity of the internal fluid will end up rotating the fluid to solid body rotation) then...
This isn't actually a homework problem, but I am still posting it here. I am confused by rotational inertia. How does mass distribution affect rotational inertia? Because I know it should, but I don't know how it would.
Homework Statement
. What is the angular acceleration if a force of 3 N is applied perpendicular to the radius of a 5 kg solid sphere that has a radius of 0.6 meters?
Homework Equations
= r x F
I = mr^2
The Attempt at a Solution
I used
tried solving for angular acce.
3N/(5kg)(0.6)...
Homework Statement
I am doing the old frozen can vs liquid can experiment and one of the questions asks that if cans with different diameters were used with the same contents (so same mass relative to diameter size) how would this effect the time taken for the cans to go down a ramp (how would...
Homework Statement
Okay, so I'm supposed to take two fresh rolls of toilet paper and drop them. One of which, I am supposed to let unravel while falling. The other, I just drop on its own. I'm supposed to find the height at which both will drop and hit the ground at the same time.
Homework...
Homework Statement
A thin rod, 0.79 m long, is pivoted such that it hangs vertically from one end. You want to hit the free end of the rod just hard enough to get the rod to swing all the way up and over the pivot.
How fast do you have to make the end go?
Homework Equations
P(change in...
If there is a disk of mass M and radius R that is already rotating, then someone puts a block with mass m on it a meters away from the center of mass.
What is the rotational inertia then?
Is it I=(M+m)R^2?
Homework Statement
>Problem:<br>Find the Moment of Inertia of a circular disk of uniform density about an axis which passes through the center and makes an angle of $\dfrac{\pi}{6}$ with the plane of the disc.
Homework Equations
Moment of Inertia ($I$) is $$\int r^2dm$$ where $r$ is the...
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...
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...
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...
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?
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...
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...
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...
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...
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...