Moment of inertia of a sphere.
The moment of inertia of the hollow sphere is 0.
Moment of inertia of a sphere Physics 12 Moment of Inertia (3 of 7) Moment of Inertia of a Hollow Sphere Michel van Biezen 1. Use our free online app Moment of Inertia of a Solid Sphere Calculator to determine all important calculations with parameters and constants. Use our free online moment of inertia calculator to compute I for rods, cylinders, spheres, plates, and rings in both metric and imperial units. p. Learn how mass distribution affects spherical objects' rotational dynamics. Apr 24, 2005 · This is not homework, but since it is related I thought it should go here. Moment of inertia of potentially tilted cuboids. 200 m has density ρ that decreases with distance r from the center of the sphere according to \rho = 3. Moment of inertia is calculated about an axis passing through its centre. 1-36. In the first example (disc method), it has the integral as dI = 1 2r2dm d I = 1 2 r 2 d m, while in the second example (shell method), dI =r2dm d I = r 2 d m. b) Calculate the moment of inertia of the sphere for an axis along the diameter. Therefore, the mass of the hollow sphere of radius r and thickness dr equals ρ 4 π r2 dr. Formula used: Theorem of parallel axis. Participants clarify the importance of distinguishing between different variables, specifically the radius of the shell and the distance from the axis of rotation. Moment of Inertia, Sphere We would like to show you a description here but the site won’t allow us. See the formula, the steps and the diagrams for both cases. Its rotational inertia is 0. The moment of inertia depends not only on the mass of an object, but also on its distribution of mass relative to the axis around which it rotates. 6836. 1) What is the centre of mass of the rest of the sphere? 2) What is the moment of inertia regarding the axis Apr 3, 2018 · I know the moment of inertia of a filled sphere is $(2/5) Mr^2$. Moment of Inertia Feb 5, 2025 · For instance, a solid cylinder’s moment of inertia is given by I = ½ (m × r 2). It is also important for understanding the dynamics of rotating objects, such as planets, stars, and gyroscopes. For example, a hollow sphere with just as much mass as a solid sphere has more rotational inertia. Perfect for physics students or anyone learning classical mechanics, this walkthrough Oct 24, 2012 · Homework Statement Homework Equations I_a = I_G + md^2The Attempt at a Solution I tried using the parallel axis theorem to find the moment of inertia about the axis. This page contains the video Moment of Inertia of a Sphere. Peter Dourmashkin Transcript Download video The mass moment of inertia measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. 29. Derivation of a moment of inertia of a sphere, just for those who are interested. The moment of inertia of a solid sphere can be used to calculate its rotational kinetic energy and angular momentum. Let be an axis in the plane of paper and a parallel axis through , the centre of mass of the body. This is version 6 of rotational inertia of a solid sphere, born on 2006-03-27, modified 2008-03-26. Jul 23, 2025 · Here , below is the derivations and formulas for the moment of inertia of common shapes—like plates, cylinders, spheres, rings, and triangles—based on their mass distribution and axis orientation, using symmetry and integration to show resistance to rotational motion. Therefore the distance you need is from a mass element in the sphere to the $z$ -axis, not to the origin. You can use 2 0 sin 3 = 2 3 This resul t can be used to calculate, by integration, the moment of inertia 2 5 m a 2 of a solid sp here. Jan 3, 2025 · The moment of inertia for a solid sphere encapsulates its rotational characteristics, depending critically on both its mass and geometric properties. Discover the derivation and calculation of the moment of inertia for a hollow sphere. My attempt: We know that MI of a Feb 3, 2008 · The discussion focuses on deriving the moment of inertia of a sphere about an axis through its center of mass, which is known to be I_ {CM} = (2/5)MR^2. Nov 24, 2017 · It did not came clear from the other answers why your approach is wrong. We review spherical coordinates and apply them to find the moment of inertia for a sphere spinning about its center of mass. 11M subscribers Subscribed The moment of inertia of a hollow sphere of mass M and radius R through any axis passing through its origin equals 2 M R2 3 (as an exercise, prove this to yourself). In this section, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object. Hollow Sphere Formula Derivation We will now understand the derivation of the moment of inertia formula for a hollow sphere. This is an incorrect image, as such a process would create a hollow sphere of much lighter mass than the uniform sphere. The moment of inertia of a solid sphere quantifies how difficult it is to change its rotational motion about an axis. The unit of dimension of the second moment of area is length to fourth power, L 4, and should not be confused with the mass moment of inertia. I hope that you enjoy the video! Please comment with any questions or suggestions Moment of Inertia, Sphere Mar 25, 2025 · Moment of Inertia is an introductory concept in rotational motion. Introduction to Classical Mechanics: With Problems and Solutions; first edition (8 January 2010). Exercise 2 6 1 A hollow sphere is of mass M, external radius a and internal radius x a. ) I'm looking at sample calculations of moment of inertia of a sphere here. Apr 29, 2025 · The moment of inertia varies with the shape and how the object rotates around its axis. Initial attempts incorrectly used the formula for thin spherical shells, leading to erroneous results. 4] Deep Dive: Gyroscopes [DD. Aug 1, 2021 · The moment of inertia is defined relative to an axis, not a point. Moment of Inertia of a Rectangular Plate If the mass of the plate is M, length l, and width b, then the moment of inertia Learn the moment of inertia of a hollow sphere with formulas, derivation, examples, and key differences vs solid spheres for JEE & NEET. The following is a list of second moments of area of some shapes. Remember that you calculate the moment of inertia for rotation around an axis not around a point. ^ A. (Take 8 = 1. Learn about its diameter, explore numerical examples, and grasp the fundamental physics principles. The solid sphere should be sliced to infinitesimally thin solid cylinders. Learn how to calculate the moment of inertia of a sphere and a thin spherical shell about their central axis. Do this by using the definition of rotational inertia for continuous masses. A lot of websites give me different solutions, so I don't know which one I have to use. m 2. Specifically, the moment of inertia of a solid sphere is directly proportional to its mass and the square of its radius. The distance between sphere and cavity center is a such that a < R-r. Clear and detailed guide on deriving the moment of inertia for a thin spherical shell. Moment of inertia is defined as a quantity that expresses the pivot energy of a particle relative to an axis, calculated as I = ∫ r² dm, where m is the mass of the particle and r is the distance from the particle to the axis. The Moment of Inertia is a quantity that defines the torque required for a preferred rotational motion around a rotational axis. Moment of Inertia, Thin Disc · Using the known value of the mass of the sphere M and calculated radius of the sphere R the moment of inertia of the given sphere about its diameter can be calculated using the given formula. Cambridge University Press. The larger the mass or the radii, the greater the moment of inertia. If Jun 5, 2016 · The moment of inertia for a solid sphere is derived as 2/5 MR^2, while the moment of inertia for a hollow sphere is 2/3 MR^2. Chalkiadakis. The tool supports calculations for: Solid Cylinder or Disc Hollow Cylinder or Thin Ring Solid Sphere Hoop Each of these shapes has a distinct moment of inertia formula based on its geometry and distribution of mass. 1M subscribers Subscribed Dec 12, 2011 · The discussion focuses on deriving the moment of inertia for a solid sphere, with initial calculations leading to an incorrect coefficient of 3/5 instead of the correct 2/5. Write the moment of inertia for a solid sphere in terms of its density rather th an its mass. Solution System, Interactions and Model: Not applicable. Mar 16, 2025 · Moment of Inertia We defined the moment of inertia I of an object to be (10. What Is the Inertia Calculator? The Inertia Calculator is an interactive tool that helps users compute the moment of inertia (I) for different objects based on mass, radius, and shape. What is the moment of inertia of this sphere about an axis through its center? 1/71 5/71 2/31 2/71 Question: Find the moment of inertia of a solid sphere of radius a about a diameter. 6. Lesson 36: Rolling Dynamics [36. In this video, we derive the formula for the moment of inertia (MOI) of a solid sphere from first principles. From what I have heard from my physics professor, it is easiest to use sphereical How to Find Moment of Inertia of Solid Sphere A uniform solid sphere of mass M and radius R is taken. We derive the moment of inertia of objects of different shapes and masses. Understanding the Moment of Inertia of a Solid Sphere One of the fundamental concepts in physics, particularly in rotational dynamics, is the moment of inertia In this article, we delve into understanding the moment of inertia of a solid sphere—a frequently encountered shape with wide-ranging applications, from engineering to astronomy. ) The moment of inertia is 0 (Type an exact answer, using a as needed. Sep 13, 2020 · The Rotational Inertia or moment of inertia of a solid sphere rotating about a diameter is This can be shown in many different ways, but here we have chosen integration in spherical coordinates to give the reader practice in this coordinate system. It plays a crucial role in analyzing spinning objects, torque, and angular momentum, making it essential for success on the AP® Physics 1 exam. Feb 11, 2025 · The moment of inertia of a solid sphere is a quantity that describes its resistance to angular acceleration. Show that x is given by the solution of 3 + 5 = and calculate x to four significant figures. Learn how to solve problems calculating the moment of inertia for a sphere and see examples that walk through sample problems step-by-step for you to improve your physics knowledge and skills. Then, according to the definition of radius of gyration we will find the radius of gyration for the sphere about that axis. 00 \times 10^3 \frac {kg} {m^3} - (9. It relies on the mass distribution of the object and the axis chosen, with bigger moments requiring more torque to affect the rate of rotation. The formula for the moment of If the object is made of a number of parts, each of whose moment of inertia is known, the total moment of inertia is the sum of the moments of inertia of the pieces. So if you choose for example the vertical axis, you notice that the points on spherical shell are at a constant distance from the center of the sphere, but they are at different distances from the vertical axis. This section is very useful for seeing how to apply a general equation to Apr 24, 2022 · Theorem 5. In the preceding section, we defined the moment of inertia but did not show how to calculate it. 62M subscribers Subscribed Moment of inertia is an essential segment in the umbrella of rotational dynamics of Physics. Because r is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis. The moment of inertia of a solid cylinder is given as I = (½)MR 2 Moment of Inertia--Sphere For a solid sphere with radius R, mass M, and density, (1) the moment of inertia tensor is (2) (3) (4) which is diagonal, and so it is in principal axis form. 5) I = I cm + m d 2 where m is the object’s mass. The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. 1) I = ∑ i m i r i 2 for all the point masses that make up the object. Mar 22, 2021 · Calculate the moment of inertia of a solid uniform hemisphere $x^2+y^2+z^2=a^2$; $z \\geq 0$ with mass $m$ about the $z$ – axis. Compared to the solid cylinder, a hollow sphere has a larger proportion of its mass located farther from the axis of rotation, so we would expect the fraction for the equation for the moment of inertia of a thin, hollow sphere about its center of mass to be more than for a solid cylinder. This is a derivation of the moment of inertia of a solid sphere, where the axis of rotation is through its center. If this seems counterintuitive, you probably carry a mental image of creating the hollow sphere by removing internal mass from the uniform sphere. The general formula for moment of inertia is given by: I = ∫ r 2 d m where r is the distance from the axis of rotation, and d m is the differential mass element. Mastering rotational inertia provides deeper insight into circular motion, real-world mechanics, and engineering Apr 7, 2015 · Physics 12 Moment of Inertia (2 of 7) Moment of Inertia of a Solid Sphere Michel van Biezen 1. ^ a b David Morin (2010). Perfect for engineering and physics students. Thus, the moment of inertia of a solid sphere equals I = R 2 ∫0 3 (ρ 4 π r2) r2 dr Mar 9, 2021 · In this system moment of inertia of the whole system about the axis shown in the question is given ( sphere is able to freely rotate about its axis) "i want to know why we are not treating sp The moment of inertia for a continuous mass distribution requires integrating the mass elements' contributions over the entire volume. Homework Equations Aug 30, 2023 · 00:00 Given the moment of inertia of a thin spherical shell, we find the moment of inertia of a solid sphere by representing the sphere with nested spherical shells then integrating thin spherical Nov 30, 2019 · I am trying to solve the Following question. Participants suggest using the method of thin disks instead, emphasizing the importance of correctly identifying the distance from the axis Jan 8, 2024 · This college physics with calculus video tutorial explains how to derive the formula for the inertia of a solid sphere. The moment of inertia for a solid sphere about an axis through its center is given by the formula: I = 2 5 M R 2 Where: I is the moment of inertia, M is the mass of the sphere, and R is the radius of the sphere. Intro to Rotational Motion: Feb 24, 2014 · A hollow sphere will have a much larger moment of inertia than a uniform sphere of the same size and the same mass. 1-37. 259mr^2, where d, the distance from the center is Oct 12, 2024 · Is it well known that the moment of inertia of a sphere can be calculated starting from its density $\rho=\frac {m} {V}$ where $V$, the volume of the solid is: $V=\frac {4} {3}\pi r^3$. Ideal for physics and engineering students. A more effective approach involves using infinitesimally The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the smaller is the angular acceleration. Object id is 142, canonical name is RotationalInertiaOfASolidSphere. Learn how to calculate the moment of inertia of a sphere about its central axis and its surface using integration and density. 320. Aug 31, 2022 · This mass moment of inertia calculator will help you determine the mass moment of inertia of any common figures like spheres, rectangular prisms, cylindrical objects, and more. ) Moment of Inertia Mar 22, 2019 · How much or how little rotational inertia an object has depends on its mass and how that mass is distributed across space. First, let us consider or recall the moment of inertia of a circle which is I = mr 2 If we apply differential analysis we get; dl = r 2 dm We have to find the dm, Feb 4, 2025 · Hollow sphere moment of inertia quantifies the resistance of a hollow sphere to rotational motion about a designated axis and depends on its mass, outer radius, inner radius, and angular velocity. The moment of inertia for any object, including spheres, is an important value calculated using the specific formulas for every object to determine the angular momentum of the entity. Or, if you start with 2 5 m a 2 for a soli d sphere, you can differentiate to find the result 2 3 m a 2 for a holl ow sphere. 3] Problem Set 12 29. Newton’s second law for rotation gives a relationship between torque, moment of inertia, and angular acceleration. It depends on the sphere’s mass, radius, and density. Oct 21, 2014 · A sphere with radius R = 0. We solve this problem using the definition of the moment of inertia. A hollow sphere is a sphere that has been thinned out to the point that a wall of equivalent thickness produces an interior Calculate The Moment Of Inertia Of A Hollow Sphere (i) About A Diameter, (ii) About A Tangent, (iii) About An Axis Through The Centre Of The Hollow Sphere. The integral is best Sep 4, 2024 · The moment of inertia of a solid sphere quantifies its resistance to rotational motion about an axis. The moment of inertia of the hollow sphere is 0. Find Moment of Inertia of a Solid Sphere Calculator at CalcTown. Approach: We will have to perform an integral over the volume of the sphere to calculate the moment of inertia. This value depends on the sphere's mass and radius and is crucial for predicting the sphere's behaviour in rotational dynamics. (Answer = 0. This section is very useful for seeing how to apply a general 00:00 Given a sphere of mass M and radius R, we derive the moment of inertia of a solid sphere about its diameter using disks. I have not yet had calculus III class, but I would like for someone to show me how the moment of inertia is derived for a solid sphere. Different objects like spheres and cylinders have specific moment of inertia formulas based on their shapes. com Nov 11, 2025 · So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk. A misunderstanding arose regarding the axis of rotation, as a teacher's derivation of 3/5 MR^2 was initially thought to be incorrect. Sep 24, 2015 · The discussion centers on calculating the moment of inertia of a sphere, which is known to be 2/5 (m) (r^2). The moment of inertia of a sphere of uniform density \rho and radius R is I = { {8\over 15}}\pi\rho R^5, so the moment of inertia of the shell created by removing a small sphere from within a big one is I = { {8\over 15}}\pi\rho (R_2^5-R_1^5). The user explores two methods: triple integration and the disk method, noting that the latter yields a doubled result due to using an incorrect formula for the moment of inertia of thin disks. In this subsection, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object. In the formula sheet they give the moment of inertia of a hemisphere: I_G = 0. The discussion clarified that the teacher likely referred to the moment of inertia about the center of the sphere, not an axis through Nov 25, 2015 · I'm trying to determine the moment of inertia of a hollow sphere, with inner radius 'a' and outer radius 'R'. This is why the disk method is used -- it takes advantage of the symmetry. For a uniform solid sphere, the moments of inertia are taken to be about an axis passing through the sphere's center of mass. Understand concepts, equations, and applications in simple terms. 5 Moment of Inertia of a Sphere Instructor:Dr. 00 \times 10^3 \frac {kg} {m^4})r a) Calculate the total mass of the sphere. Nov 10, 2018 · Deriving moment of inertia of a solid sphere [closed] Ask Question Asked 7 years ago Modified 7 years ago • Use the moment of inertia equations to determine how each variable effects the ease of rotation for a hollow hoop, solid cylinder, and sphere. To see this, let’s take a simple example of two masses at the end of a Learn the moment of inertia of a sphere with detailed derivation, explanation, and formulas for both solid and hollow spheres. But there is an additional twist. 3. Furthermore, it is inversely proportional to the sphere's density, offering insights into its material composition A uniform solid sphere has a moment of inertia I about an axis tangent to its surface. The formula for calculating the moment of inertia of a solid sphere and hollow sphere is derived below in the blog. Another example is the moment of inertia for a solid sphere, which is given by I = ⅖ (m × r 2), where r represents the radius of the sphere. 5 M a 2. Jan 25, 2025 · Discover the rotational inertia of a sphere, including moment of inertia, torque, and angular momentum, to understand its rotational motion and kinetic energy. 1-DD. How do you show the moment of inertia of a hollow sphere is $(2/3) Mr^2$? Apr 10, 2024 · In the preceding subsection, we defined the moment of inertia but did not show how to calculate it. Find all the components of moment of inertia I have the following approach to solve this. The general formula calculates moment of inertia using distance and mass of particles rotating together. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis. See full list on miniphysics. The moment of inertia of a body about any axis is equal to the sum of its moment of inertias about a parallel axis through its centre of gravity and the product of its mass and the square of the distance between the two axes. Consider a sphere of radius R with a cavity of radius r cut out of it. The correct In the preceding section, we defined the moment of inertia but did not show how to calculate it. ISBN 978-0521876223. Now I know that we will use parallel axis theorem between axis through center of cavity and through center of sphere, BUT I had first tried to assume that both their axes are NOT parallel, but the same. It is not examinable in PHYS1121/1131/1141, the formula is on your formula sheet. The moment of inertia of a solid sphere is vital from a 3-D perspective. Mar 7, 2025 · Rotational inertia is a fundamental concept in rotational motion, describing an object’s resistance to changes in its angular velocity. AI generated definition based on: Divided Solids So I need to find the remaining portion's Moment of Inertia about the sphere's diameter. Technical report, University of Southampton, 2015. Nov 24, 2009 · Calculate the moment of inertia of a solid sphere of radius R and uniform density ρ rotated about an axis passing through its center. 5 Deep Dive - Moment of Inertia of a Sphere MIT OpenCourseWare 5. I R = I C O M + M R 2 Complete answer I have been given a sphere of radius a, from this sphere a cap of hight h is cut off. Panagopoulos and G. It depends on both the mass of the sphere and the distribution of that mass relative to the axis of rotation, influencing how a solid sphere responds to applied torques during rotational dynamics. Furthermore, because of the symmetry of the sphere, each principal moment is the same, so the moment of inertia of the sphere taken about any diameter is The Moment of Inertia Calculator is an essential online physics tool that helps you quickly determine the rotational inertia of different objects such as spheres, cylinders, and rods. 5] Lesson 37: Rolling Kinetic Energy & Angular Momentum [37. This section is very useful for seeing how to apply a general equation to Jul 2, 2019 · The moment of inertia of the hemisphere about an axis parallel to O O passing through a a (the "base" of the hemisphere) is ICoM = 2 5ma2 I C o M = 2 5 m a 2 Therefore, from the parallel axis theorem, IO = 2 5ma2 +ma2 = 7 5ma2 I O = 2 5 m a 2 + m a 2 = 7 5 m a 2 The general expression for the moment of inertia is:. 4. 528 kg. Aug 2, 2023 · The moment of inertia of a rotating object about a fixed axis is useful in calculating a few key quantities in rotational motion. Find the moment of inertia of a uniform solid sphere of radius 𝑎 and mass 𝑚 about an axis which lies at a distance √ (2𝑎)/2 from the center of the sphere. Difficult Derive the equation for the rotational inertia of a sphere of mass m and radius r about one of its internal axes. 1: Parallel axis theorem If the moment of inertia of a rigid body about an axis through its center of mass is given by I c m, then the moment of inertia around an axis parallel to the original axis and separated from it by a distance d is given by (5. Type an expression using a as the variable. Hint: First we will be using the theorem of parallel axis to find the moment of inertia about the axis parallel to an axis passing through the centre of the sphere and tangent to the sphere. Other shapes have their own unique formulas for calculating moment of inertia. See the expressions, derivations, and examples for both cases. This calculator simplifies the process of finding how mass and shape affect an object’s resistance to rotational motion — a crucial concept in mechanics, engineering, and rotational dynamics. For a sphere, it can be specifically defined as I = (2/5)mR², where m is the mass and R is the radius of the sphere. 5 Moment of Inertia of a Sphere 29. gzjfndofhsaxedbsrzoueskqvxokzelxdzdyutzrgnijgsonwmbfzbhvhltyzglysjm