# What is Parallel axis theorem: Definition and 57 Discussions

The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

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1. ### Angular velocity of a loop after being struck by bullet

First off, I do know how to solve this problem. We use the principle of conservation of angular momentum about the centre of mass of the system which comprises of the loop and the bullet to obtain option B. My doubt is, why do we just not use the principle about the centre of the loop? Where is...
2. ### Parallel Axis Theorem Not Working

Using the trivial way, you get I = M(r+L)^2. This is definitely correct. Out of curiosity, can the parallel axis theorem be applied here? I took the axis of rotation to be into the page, then calculated the moment of inertia for the disk, which is I = 1/2Mr^2. Next, I calculated the distance...
3. ### I Explanation of parallel axis theorem

For a rotating system with mass m this theorem says that if it rotates about an axis distance x from but parallel to the axis through it's natural mass center (CM), then I moment of inertia is $$I=I_{CM}+mx^2$$ My thinking is if one move the axis x distance away from the axis through it's CM...
4. ### I Resolve moment of inertia at an angle

Initially, I calculate the moment of inertia of of a square lamina (x-z plane). Thr this square is rotated an angle $\theta$ about a vertex and I need to calculate the new moment of inertia about that vertex. Can I split the rotated square to two squares in the x-z plane and y-z plane to find...
5. ### Determine the moment of inertia of a bar and disk assembly

I have been given an answer for this but I am struggling to get to that point $$ANS = 0.430\, kg \cdot m^2$$ So I thought using the moment of inertia of a compound pendulum might work where ##I_{rod} = \frac{ml^2}{12}## and ##I_{disc} = \frac{mR^2}{2}## (##l## is the length of the rod and ##R##...
6. ### Proof related to the center of mass

hi guys in the proof of the parallel axis theorem this equation is just put as it is as a definition of the center of mass : $$\int[2(\vec{r_{o}}.\vec{r'})I-(\vec{r_{o}}⊗\vec{r'}+\vec{r'}⊗\vec{r_{0}})]dm = 0$$ is there is any proof for this definition ? and what is the approach for it
7. ### B When is the parallel axis theorem not appliable?

Hello, there. A friend asked me a problem last night. Suppose that a system consists of a rod of length ##l## and mass ##m##, and a disk of radius ##R##. The mass of the disk is negligible. Now the system is rotating around an axis in the center of the disk and perpendicular to the plane where...
8. ### Understanding Feynman's Discussion of the Parallel Axis Theorem

https://www.feynmanlectures.caltech.edu/I_19.html "Suppose we have an object, and we want to find its moment of inertia around some axis. That means we want the inertia needed to carry it by rotation about that axis. Now if we support the object on pivots at the center of mass, so that the...
9. ### Why Use the Parallel Axis Theorem with the Second Moment of AREA?

Hi all I was wondering if someone could help clear up some confusion about the Parallel Axis Theorem. I am trying to understand the purpose/benefit of applying the Parallel Axis Theorem with respect too the Second Moment Of Area. For example I have a beam that is under load. I have found its...
10. ### Where does the plane exist in this parallel axis theorem?

##I_{AB} = I_{GXX} + A.(y^{2})## Same applies to CD; ##I_{CD} = I_{GYY} + A.(x^{2})## In the above statement, "any axis in its plane" where does the plane exist in this sketch?
11. ### Parallel Axis Theorem / Bending Stress

Homework Statement find bending stress in x and y dir Homework Equations I = bh^3/12 + ad^2 Stress = Mc/I The Attempt at a Solution I = bh^3/12 + ad^2 Stress = Mc/I see attached calculations My prof gave us a question where we have a motor (20" tall) sitting on a frame with a load of...
12. ### Parallel Axis Theorem Experiment

Homework Statement I am currently working on a physics experiment to confirm the parallel axis theorem. To do this, I have the following setup: In this experiment I change the distance between the centre of the rotating disc and the central axis. I record the time for 5 complete rotations...
13. ### Kinetic energy of rotation and parallel axis theorem problem

Homework Statement A circular disc of radius 25 cm and mass 0.5 kg is revolving in its plane with an angular velocity of 4 radians per second. Find A) its kinetic energy of rotation, and B) its new angular velocity if a mass of 10 kg is suddenly fixed on the rim of the disc. Homework...
14. ### Parallel Axis Theorem- Composite Areas (STATICS)

Homework Statement Consider the beam shown in (Figure 1) . Suppose that a = 15 in. , b = 8 in. , c = 1 in., and d = 4 in. Determine the moment of inertia for the beam's cross-sectional area about the x axis...
15. ### Distance between 2 axis in Parallel Axis Theorem

Homework Statement The moment of inertia for a perpendicular axis through the center of a uniform, thin, rectangular metal sheet with sides a and b is (1/12)M(a2 + b2). What is the moment of inertia if the axis is through a corner? The answer is given as this was a powerpoint lecture and it...
16. ### Why is Icm not needed in the Parallel Axis Theorem?

Homework Statement Homework Equations Parallel axis theorem: Ip = Icm + Md^2 Icm = I = ML²/12 + 2 * mr² 3. The attempt Ip = Icm + Md^2 ==> wrong I = Md^2 ==> right Why don't I need to add "Icm"? Thanks.
17. ### Applying the parallel axis theorem to find inertia

Homework Statement Calculate the moment of inertia of a uniform rigid rod of length L and mass M, about an axis perpendicular to the rod through one end. Homework Equations Parallel axis theorem: I = Icm + MD2 Long thin rod with rotation axis through centre: Icm = 1/12 ML2 Long thin rod with...
18. ### When to use parallel axis theorem for objects....

Homework Statement A uniform solid ball of mass m and radius R rolls without slipping down a plane inclined at an angle f above the horizontal. Find the frictional force and the acceleration of the center of mass.[/B]Homework Equations τ=I*α so: fs*r=I*a Mg-Fs=ma Moment of inertia for...
19. ### Period of small oscillations for a pendulum

Homework Statement A pendulum consists of a light rigid rod of length 250 mm, with two identical uniform solid spheres of radius of radius 50 mm attached one on either side of its lower end. Find the period of small oscillations (a) perpendicular to the line of centres and (b) along...
20. ### Parallel axis theorem for area

Homework Statement why the y bar is 0 ? according to the diagram , y ' has certain value , it's not 0 ! can someone help to explain ? Homework EquationsThe Attempt at a Solution
21. ### Do I understand this correctly?

Homework Statement the theory behind Parallel axis theorem Homework Equations parallel axis theorem:Io=Ig+md² Radius of Gyration=Ig=m*K² The Attempt at a Solution ok folks If I understand this theory correctly the radius of gyration is the radius or distance from the axis of rotation to the...
22. ### What is the definition of the D-Offset in the Parallel Axis Theorem?

Homework Statement My question is, to gain more knowledge on, is in Physics, what is the terminology for D-offset in the Parallel axis theorem? Homework Equations I= Icm + Md^2 The Attempt at a Solution From my understanding, the offset is the distance away from the axis of rotation.
23. ### Moments of inertia using parallel axis theorem

Homework Statement The mass of a homogenous thin plate is 36kg. Determine it moment of inertia about the x axis Homework Equations Parallel axis theorem: Ix = Ix' + d^2*m The Attempt at a Solution p = m/a = 36/0.36 = 100 Because the density is homogenous, I found the mass of the first...
24. ### Can someone please explain what the parallel axis theorem is?

Can someone please explain what the parallel axis theorem is?
25. ### What is the moment of inertia and angular velocity of a released rod?

A uniform rod of mass M=5.0 kg and length ℓ=20 cm is pivoted on a frictionless hinge at the end of it. The rod is held horizontally and then released. a) Use the parallel-axis theorem to determine the moment of inertia of the rod about the hinge (ie its end). b) Determine the angular...
26. ### Parallel Axis Theorem derived from the Displaced Axes Theorem.

I am trying to attain the parallel axis theorem from the displaced axes therom. I have the displaced axes thorem stated in this form: \hat{I}=\hat{I}com+M\hat{A} -Where Rc is the position of the centre of mass position- -Where \hat{}is the inertia tensor of a rigid body wrt to rotations...
27. ### Parallel Axis Theorem problem help

Homework Statement This is moreof a conceptual question regarding the parallel axis theorem and area. Lets say I have a solid rectangle with length L, and Width W. I cut out a circle of radius R at the center. When calculating the Total Area moment of inertia of the hollow region where the...
28. ### Proof of parallel axis theorem.

Everybody says that the distance ,between the two axis, used in the formula, is perpendicular. But in the proof it was a hypotenuse. It was not perpendicular.
29. ### Moment of Inertia Using Parallel Axis Theorem

Homework Statement Q is the edge length of the cube. The moment of inertia of the cube about an axis passing through its center and the center of two opposing faces is (1/6)mQ^2 (PCM in the diagram). Use the parallel axis theorem to find the moment of inertia if the axis is along one face of...
30. ### How Do You Apply the Parallel Axis Theorem to Calculate Moment of Inertia?

Homework Statement Homework Equations I=Bmr^2 Parallel axis theorem = Icm + MD^2 WET, KE, PE equations The Attempt at a Solution So far I've only done parts a and b and I wanted to post this up as soon as possible, I want to make sure if I'm on the right path so far. part...
31. ### Parallel Axis Theorem Clarification

in what situations would you require the use of the parallel axis theorem? Also, from the physics book it says that let x and y coordinates of P(a point parallel to the first axis) be a and b. then let dm be a mass element(what does this mean? a point anywhere within the object?) with the...
32. ### Parallel Axis Theorem | Kinetic and Potential Energy

Homework Statement A rigid body is made of three identical rods, each with length L = 0.525 m, fastened together in the form of a letter H, as in Figure 10-56 below. 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 is allowed...
33. ### Inertia with the parallel axis theorem

Homework Statement A uniform solid sphere has a moment of inertia I about an axis tangent to its surface. What is the moment of inertia of this sphere about an axis through its center? Homework Equations Ip = Icm + Md2 Isolid sphere = 2/5MR^2 The Attempt at a Solution This is...
34. ### Proof of Parallel Axis Theorem

Note: The following are taken from Physics for Scientists and Engineers 6E http://img542.imageshack.us/img542/821/75796098.png http://img152.imageshack.us/img152/5615/70724407.png http://img804.imageshack.us/img804/6813/65815357.png I don't really understand why is that integral...
35. ### Rotational Motion- Parallel Axis Theorem?

Homework Statement A meter stick is held to a wall by a nail passing through the 60-cm mark. The meter stick is free to swing about this nail, without friction. If the meter stick is released from an initial horizontal position, what angular velocity will it attain when it swings through the...
36. ### What is and how do I use the Parallel axis theorem

I have in my problem, a ball sitting on a cylindrical rod that pivots at the bottom, some guy in the explanation said parallel axis theorem and came up with moment of inertia of: I = (mL^2)/3 + [2Mr^2/5 + M(L+r)^2] where L is the length of the rod, m is the mass of the rod, M is the mass of the...
37. ### Parallel axis theorem, cube (Confirm)

Parallel axis theorem, cube! (Confirm) // Idisp = Icenter + mass[ (RdotR)*I - RcrossR ] So to test it out, I create a long box at the origin, and then a smaller box, half its width, so I can offset it along the x axis, and times it by 2, so it should equal the inertia tensor of the long...
38. ### Stress Analysis - Sections (inc. Parallel axis theorem)

im currently stuck on a stress analysis question, the question is the following: for the section shown below, derive the following: x, y, Ixx, Iyy, Ixy, theta, Iu and Iv 50 ___ yy |---| | |---| | |---| | |---| +------ xx |---|___________________...
39. ### Help with parallel axis theorem?

help with parallel axis theorem?? Hey guys, I've attached a picture from my textbook (Intro to Classical Mechanics by David Morin) showing the beginning of the proof for the parallel axis theorem. I understand most of it except the sentence where it states that if you glue a stick to the body...
40. ### How to use parallel axis theorem?

Homework Statement A solid door of mass 37.80 kg is 2.30 m high, 1.70 m wide, and 2.53 cm thick. What is the moment of inertia of the door about the axis through its hinges? Homework Equations I= Icm + MD^2 Icm = 1/12[M (a^2 +b^2 )] formula for inertia of a rectangular plate. The...
41. ### How Do You Apply the Parallel Axis Theorem to Calculate Moment of Inertia?

Hi everyone, I've got stuck on this prove problem:cry: Please help me! Let S be a rectangular sheet with sides a and b and uniform density, and total mass M. (a) Show that the moment of inertia of S about an axis L that is perpendicular to S, meeting S through its center, is I...
42. ### Why do I get a different answer with the parallel axis theorem?

Why do I get a different answer with the parallel axis theorem? [Solved] Homework Statement Imagine four points masses m1 = m3 = 3kg and m2 = m4 = 4 kg. They lie in the xy plane with m1 at the origin, m3 at (0, 2), m2 at (2, 2), and m4 and (2, 0). Each unit on the coordinate plane...
43. ### Moment of inertia and Parallel axis theorem

1. Thehttp://nft01.nuk.edu.tw/lib/exam/97/master/97ap-master.pdf" statement, all variables and given/known data A solid, uniform disk of mass M and radius R is oscillating about an axis through P. The axis is perpendicular to the plane of the disk. Suppose the friction at P can be ignored. The...
44. ### How Is the Parallel Axis Theorem Applied in Physics IA?

Homework Statement Prove parallel axis theorem (Steiner's theorem). Homework Equations I_{parallel axis} = I_{cm} + Mr^2 The Attempt at a Solution I know Wolfram's site which seems to use http://scienceworld.wolfram.com/physics/ParallelAxisTheorem.html" I am not sure whether the hint of...
45. ### Help with proving the parallel axis theorem

Hi, this is my first post here, so please forgive any errors in my post. :) I'm recently thinking about switching from my major of computer science to physics, and have been brushing up on the first few semesters of physics I had taken a few years ago. I'm currently in the section on...
46. ### Parallel axis theorem + rotational inertia

Homework Statement Two particles, each with mass m = 0.85 kg, are fastened to each other, and to a rotation axis at O, by two thin rods, each with length d = 5.6 cm and mass M = 1.2 kg. The combination rotates around the rotation axis with angular speed w = 0.30 rad/s. Measured about O, what...
47. ### Parallel Axis Theorem and interia tensors

Is the parallel axis theorem always valid for inertia tensors? We have only seen examples with flat (2d) objects and was wondering if it would also be valid for 3d objects, like a h emisphere, for example. Thanks.
48. ### Parallel Axis Theorem and sheet of metal

Homework Statement A thin, rectangular sheet of metal has a mass M and sides of length a and b. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet that passes through one corner of the sheet Homework...
49. ### Moment of Inertia and the parallel axis theorem

Homework Statement I have a problem that involves a vertical axis that can rotate through a fixed axis located at the very top of the rod, at point A (the axis is coming out towards us, the rod's rotation would look like a pendulum). The rod is nonuniform and there is a block attached to the...
50. ### Explain to me the parallel axis theorem

hi there. can anyone please explain to me the parallel axis theorem? the parallel axis theorem states that I = I_cm + M(d^2) where d = distance from the center of mass axis to the parallel axis and M is the total mass of the object. The rotational inertia of a thin rod about the center is =...