What is Moment of inertia: Definition and 1000 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. M

    Moment of Inertia: Difference Between Equations 3 and 6

    What is the differece between the two formulas numbered 3 and 6 in the paint document? And what types of questions would I use eq. 3 and eq. 6?
  2. B

    Moment of Inertia: A Fundamental Property of Rotating Objects

    What is "moment of inertia?" Just curious and I use it a lot but I am not entirely sure what it is. Call me an idiot but I need to know before my physics endeavors proceed.
  3. B

    What is the derivative of the moment of inertia?

    This just struck me and I was curious.
  4. T

    Calculating new moment of inertia matrix with a new center of mass

    Homework Statement The total length of the composite body is 4.5 feet. Before the propellant is burned, the projectile weighs 23 lbf. After the propellant burns, the remaining projectile weighs 16 lbf. Before the propellant is burned, the mass center is located 2.6 feet from the projectile...
  5. A

    The moment of inertia of circular sector

    Hello how can I find the moment of inertia of a circular sector about the X axis , which the sector is symmetrical about , with -θ down and θ above ?! I_{x} = ∫y^{2} dA = ∫y^{2} *y *dx Or =∫∫ y^{2} dy dx I don't know how to put the limits of integration , I turned it to polar...
  6. B

    How Do You Calculate Angular Acceleration for a Rotating Beam?

    Homework Statement A uniform beam of mass m = 0.6 kg and length L = 0.3 m can rotate about an axle through its center. Four forces are acting on it as shown in the figure. Their magnitudes are F1 = 1.5 N, F2 = 1.5 N, F3 = 1.5 N and F4 = 1.5 N. F2 acts a distance d = 0.12 m from the center of...
  7. P

    Moment of inertia of disk, the easy way out?

    Homework Statement When calculating moment of inertia of a disk there is something that really bothers me. I've googled this a lot and everywhere i look they 'assume' that the Δa = Δr*2∏r, formula for rectangle, not circle: (area of circle r+Δr - area of circle r) Δa = ∏(r+Δr)^2 - ∏r^2 = ∏r^2 +...
  8. U

    How Does Water Rotation Affect Moment of Inertia in a Cylindrical Tube?

    Homework Statement A closed cylindrical tube containing some water (not filling the entire tube) lies in a horizontal plane. If the tube is rotated about a perpendicular bisector, the moment of inertia of water about the axis a)increases b)decreases c)remains constant d)increases if the...
  9. B

    Rotational kinetic energy / moment of inertia

    Homework Statement A thin, cylindrical rod = 27.0 cm long with a mass m = 1.20 kg has a ball of diameter d = 10.00 cm and mass M = 2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in the figure below. The combination is...
  10. L

    Moment of inertia rolling down a hill

    Homework Statement Two solid spheres -- a large, massive sphere and a small sphere with low mass -- are rolled down a hill. Which one reaches the bottom of the hill first? Homework Equations ICM= (2/5)MR2 The Attempt at a Solution I thought that this would be the smaller sphere...
  11. L

    Designing a Car for Coasting Race: Wheels

    Homework Statement Suppose you are designing a car for a coasting race -- the cars in this race have no engines, they simply coast down a hill. Do you want large wheels or small wheels? Do you want solid, disk-like wheels, or hoop-like wheels? Should be wheels be heavy or light? (Select all...
  12. T

    Moment of Inertia of Half a Sphere

    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. In the formula sheet they give the moment of inertia of a hemisphere: I_G = 0.259mr^2, where d, the distance from the center is...
  13. X

    Moment of inertia of a triangular plate

    Homework Statement see attachment Homework Equations integration The Attempt at a Solution answers: a. mH2/6 + mt2/12 b. mB2/2 + mt2/12 c. mH2/6 + mB2/2 d. -mBH/4 e. 0, 0 If the plate were thin t can be ignored. Ok so e is because of symmetry so I get that. a-d on the...
  14. S

    Moment of Inertia for a uniform wire

    Homework Statement A flat object shown here consists of a circle and square made of heavy, uniform wire and welded together at the corners of the square. The mass of the circle is M. The mass of the square is m and its side has a length d. To find the objects moment of inertia about Axis A...
  15. C

    Moment of inertia and rotational kinetic energy

    I can't seem to understand moment of inertia. What does it mean and how is it derived ? How does it relate to rotational kinetic energy.
  16. S

    Moment of inertia spinning disk integration

    Homework Statement I want to calculate the moment of inertia of a spinning disk via integration. I'm aware of the perpedicular axis theorem, but I want to integrate. Homework Equations I = ∫r^2dm The Attempt at a Solution if I set my coordinate axis op so that the origin of the...
  17. R

    Moment of Inertia? Potential and Kinetic Energy? Anybody awake?

    I don't know how to answer these - please please please help! Consider the following objects of mass m rolling down an incline of height h. (a) A hoop has a moment of inertia I = (1/2)mr2. What is the equation for the velocity vhoop of the hoop at the bottom of the incline? (Use the...
  18. X

    Moment of inertia (integration)

    Homework Statement See attachment Homework Equations Integrating The Attempt at a Solution The answer is 16pi slug-ft^2. Now I know your suppose to integrate it. Its in 2D, so a double integral at most, maybe a single integral? I'm not very good with integrals and their limits...
  19. M

    Calculating the Moment of Inertia of an Irregular Rod Shape

    To give you a better idea, I have it drawn out here: http://tinypic.com/r/eq6ln5/6 I am calling the thickness of a rod t and the thickness of the shaft t2. I am using the basic equation Ixx = (integrate over area)(y^2)dA on different sections and then adding them all together, following the...
  20. A

    Moment of inertia: off-central diameter

    Question for the brilliant minds around here: I'm trying to figure the amount of torque (I*a) needed to rotate a cylinder of a given mass, diameter and length around a pivot point that is off-center. Typically I'm assuming I would find it out by calculating both I for central diameter and end...
  21. B

    How to Calculate Moment of Inertia for a Solid Disk with Central Hole

    Consider a solid disk made of aluminum with a central hole as shown in the figure - can't include...don't believe it's necessary. The external and internal diameters are found to be 13 inches and 0.6 inches. The disk is 0.5 inch thick. The density of aluminum is 2.70 g/cm3. Question: Calculate...
  22. L

    Homework Help: Calculating Moment of Inertia & Radius of Gyration

    Homework Statement Homework Equations radius of gyration: r = root (I/m) I = moment of inertia m = mass parallel axis theorem given above The Attempt at a Solution Okay, so I think the moment about CM is just m*0.24^2, but after that, I'm less sure. Is the moment about the hip just...
  23. H

    Moment of Inertia of a uniform triangle(involves integration)

    Homework Statement The attachment shows an equilateral triangle of side length "2d" It is a uniform triangle in the 2-D space. Mass of triangle = M I have to find the moment of inertia about one of its side . I am taking the side \overline{AB} as the axis of rotation (hence i would be...
  24. N

    Moment of Inertia of a Torus

    Homework Statement To calculate the moment of inertia of a solid torus through the z axis(the torus is on the xy plane), using the parallel and perpendicular axis theorem. Homework Equations The Attempt at a Solution Well, first I divided the torus into tiny little disks and...
  25. M

    Is it possible to directly compute the maximum moment of inertia for an object

    Given some arbitrary shape I can compute the moment of inertia about any axis without a problem by summing the inertia of each of the shapes making up the entire object. I also know the center of mass of the object. Is it possible to directly compute the angles of the axis for the minimum and...
  26. D

    Moment of inertia of a few basic objects

    Ok i need some help with some homework that is to derive formula for moment of inertia of a few objects about the axis's that i have mentioned 1. Rectangular slab about axis through center(sides a,b) 2. Annular cylinder about central axis (radii R1 and R2) The only equation i know is...
  27. mesa

    Moment of inertia question; tube, solid cylinder on inclined plane.

    If you have a tube and a solid cylinder of the same dimensions and density and rolled them down an inclined plane the 'tube' would cover the same distance in less time?
  28. E

    Moment of Inertia of a body with linearly increasing density?

    A thin wire of length L and uniformly density ρ is bent into a circular loop with center at O.The moment of inertia of it about a tangential axis lying in the plane of loop is. Ans : Mass M is not given,but ρ is given. So M=ρL3->(1) (L3 means L cube,no idea how to post it in that manner!). For...
  29. E

    Moment Of Inertia of broken disk or ring confusion

    We all know that M.I of a Uniform rigid rod about an axis perpendicular to it's length and passing through it's center is MLsquare/12.Where M is mass and L is length of the rod. If it is broken to half such that M becomes M/2 and L becomes L/2,we can't apply ML square /12 formula to it.We have...
  30. B

    Moment of inertia and force needed to tilt/change axis of rotation

    Consider a freely rotating body. Let the axis of rotation be the z-axis. For simplicity assume all the mass of the body is concentrated in the x-y-plane, i.e. the plane in which the body rotates. I have read about the moment of inertia tensor on wikipedia, but I don't see how I would combine...
  31. W

    The moment of inertia of a group of seven pennies

    Homework Statement See attached photo Homework Equations The Attempt at a Solution I figured I would use the parallel axis theorem. I'm stuck between two different methods of doing the question, both of which are choices in the answers. My gut instinct says to take the...
  32. U

    Moment of Inertia of Disc through diameter

    Homework Statement The problem is attached in the picture. The Attempt at a Solution I managed to solve it using a different method. I have no idea what the answer is talking about.. My method Found dI of a strip = (1/3)*dm*h2 then i replace h by x, then integrate from -a to...
  33. P

    Effect of moment of inertia on rolling distance

    Hi everyone, good day. this might be a simple question, but I need someone to check my answer. A disk and a hoop, of same mass and same diameter, is first giving a torque (same amount of torque for both) then the torque is removed (the torque is acting on them for the exact same period of...
  34. C

    Conservation of Angular Momentum Experiment: Moment of Inertia

    Homework Statement I did a lab where there was a rotating solid disk with mass= 0.915kg and diameter=0.253m. This was rotating horizontally with an initial angular velocity of 3 different values ω radians/second. After recording the initial angular velocity, I dropped a thin-walled hollow...
  35. R

    Moment Of Inertia (calc vs physics different answer)

    I'm trying to understand Moment Of Inertia using integration. But it seems the calculus definition of physics definition are different. (I'm trying to apply my math skill to physics) example of apparent contradiction: here:http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#mi it says the...
  36. P

    Moment of inertia of system in 3D

    Hey i am working on something and i need to know how to calculate moment of inertia of a 3D system of objects. I know these variables: Mass of whole system Center of mass of whole system Center of mass of each object Offset of each object Mass of each object Moment of Inertia of each...
  37. A

    Moment of inertia for ball rolling up a ramp.

    Homework Statement A ball with mass 1.0 kg and radius 0.20m rolls without slipping along level ground with a speed of 10 m/s. The ball then rolls up an incline reaching a maximum vertical height of 8.0 m. What is the moment of inertia of the ball? (Do not assume the ball is a uniform sphere)...
  38. C

    How Do You Calculate the Second Moment of Inertia for a Reinforced Hollow Tube?

    Hello, I am trying to calculate the second moment of inertia for a hollow tube with a reinforcement bar that goes through the tube. The cross section is basically a thin-walled circle (0.1" thickness) with a horizontal bar (about 0.1" thick for now) that spans the diameter of the circle. How...
  39. S

    Find the moment of inertia of a solid sphere.

    Homework Statement Beginning with Icm = Integral of r^2 dm from r1 to r2, find the moment of inertia of a solid sphere about any tangential axis. Homework Equations Icm = Integral of r^2 dm The Attempt at a Solution I set up the infinitesimally mass of an infinitesimally...
  40. P

    Moment of Inertia for a Square and a pipe of variable density

    Homework Statement First, there's a slender rod with length L that has a mass per unit length that varies with distance from the left end, where x=0, according to dm/dx = yx where y has units of kg/m^2. (a) Calculate the total mass of the rod in terms of y and L (Which I've already done and...
  41. A

    Find torque in the x direction given moment of inertia tensor for plate.

    This is for prelim study. Just wondering if this solution is correct. Problem A thin homogeneous plate lies in the x-y plane. Its moment of inertia tensor in the x,y,z basis is given by \textbf{I}=σl^{4}\begin{pmatrix} 2 & -2 &0 \\ -1 & 2 & 0 \\ 0 & 0 & 4\end{pmatrix} If the plate...
  42. D

    Moment of inertia of a uniform solid sphere

    For calculating I of a uniform solid sphere, why can't we use thin spherical shells? When I try to use spherical shells I get (3/5)MR^2. Every single derivation uses thin cylindrical shells and end up with the correct expression((2/5)MR^2) but they never explain why it is correct to use...
  43. Avatrin

    Moment of inertia of a three particle system

    Homework Statement I have three particles of mass m at (-a,-a), (a,-a) and (0,a) Find the moment of inertia I_z around the center of mass of the system for an axis along the z-axis. Homework Equations Center of mass: CM = Ʃmr Moment of inertia: I = Ʃmp^2 m = mass r = position...
  44. N

    Moment of Inertia of a Hollow Cylinder

    Homework Statement A hollow cylinder has mass m, an outside radius R2, and an inside radius R1. Use integration to show that the moment of inertia about its axis is given by I = 1/2*m(R2^2 + R1^2) Homework Equations dm = rho*dV = 2*pi*rho*h*r*dr The Attempt at a Solution This...
  45. A

    Calculating Rotor Moment of Inertia and Time to Slow Down

    Ok, here we go. In a rotating mechanism (helicopter rotor), at a state of equilibrium, the rotor consumes a certain amount of energy from the shaft to maintain a constant angular velocity (since there is a measure of resistance present over the span of the rotor). Lets suppose that the...
  46. M

    Measuring Angular Momentum Changes in a Rotational Inertia Experiment

    In my experiment based on the inertia the angular velocity or speed goes either up or down with the angular momentum remaining the same. I understand that when a person is rotating to change the inertia you would either extend or detract the arms and legs (think dancer). I understand this...
  47. C

    What is the Moment of Inertia of a Circular Thin Cylindrical Surface?

    Homework Statement Find the moment of inertia of a circular thin cylindrical surface which ranges from -α/2 to α/2. So looks like - ) The dash being the origin. It basically looks like one fifth of a circular ring. Homework Equations I = mr² The Attempt at a Solution...
  48. L

    Moment of inertia formula for a propeller

    Is there a moment of inertia formula for a 2 bladed propeller? If you only have the mass and the length of the propeller, I would think a good estimate would be using a rod rotating about the middle. 1/12ml2
  49. L

    Area moment of inertia of inverted triangle?

    Homework Statement A inverted isosceles triangle gate with height a=3ft and base b=2ft is 6ft under the water (top of the inverted triangle). Find the Force on the gate and hp (the depth of the) center of pressure. Homework Equations hc= depth to gate + depth to centroid= 6+(1/3)•3= 6+1 =...
  50. S

    Moment of inertia: vector derivation

    We have the representation of torque attached. The components of r are (x,y,z). Where did the matrix come from and how did we get the stuff in the matrix? (Basically I understand all the steps except the step from the 3rd line to the 4th line.) Thank you very much!
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