Use Coupon: CART20 and get 20% off on all online Study Material

Total Price: R

There are no items in this cart.
Continue Shopping
Get instant 20% OFF on Online Material.
coupon code: MOB20 | View Course list

  • Complete Physics Course - Class 11
  • OFFERED PRICE: R 2,800
  • View Details
Get extra R 700 off



6 years ago


Answers : (1)



Tensors, defined mathematically, are simply arrays of numbers, or functions, that transform according to certain rules under a change of coordinates. In physics, tensors characterize the properties of a physical system, as is best illustrated by giving some examples (below).

A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points. In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. This just means that the tensor is defined at every point within a region of space (or space-time), rather than just at a point, or collection of isolated points.

A tensor may consist of a single number, in which case it is referred to as a tensor of order zero, or simply a scalar. For reasons which will become apparent, a scalar may be thought of as an array of dimension zero (same as the order of the tensor).


In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, or the angular mass, (SI units kg·m2) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.

While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.


Moment of inertia tensor

In three dimensions, if the axis of rotation is not given, we need to be able to generalize the scalar moment of inertia to a quantity that allows us to compute a moment of inertia about arbitrary axes. This quantity is as the moment of inertia tensor and can be represented as a symmetric positive semi-definite matrix, I. This representation elegantly generalizes the scalar case: The angular momentum vector, is related to the rotation velocity vector, ω by


and the kinetic energy is given by

\frac{1}{2} \omega^\top \mathbf{I} \omega

as compared with

\frac{1}{2} I\omega^2

in the scalar case.

Like the scalar moment of inertia, the moment of inertia tensor may be calculated with respect to any point in space, but for practical purposes, the center of mass is almost always used.


For a rigid object of N point masses mk, the moment of inertia tensor is given by

 \mathbf{I} = \begin{bmatrix} I_{11} & I_{12} & I_{13} \\ I_{21} & I_{22} & I_{23} \\ I_{31} & I_{32} & I_{33} \end{bmatrix} ,


I_{11} = I_{xx} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (y_{k}^{2}+z_{k}^{2}),\,\!
I_{22} = I_{yy} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (x_{k}^{2}+z_{k}^{2}),\,\!
I_{33} = I_{zz} \ \stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} (x_{k}^{2}+y_{k}^{2}),\,\!
I_{12} = I_{xy} \ \stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} x_{k} y_{k},\,\!
I_{13} = I_{xz} \ \stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} x_{k} z_{k},\,\!
I_{23} = I_{yz} \ \stackrel{\mathrm{def}}{=}\  -\sum_{k=1}^{N} m_{k} y_{k} z_{k},\,\!

and I12 = I21, I13 = I31, and I23 = I32. (Thus I is a symmetric tensor.)

Here Ixx denotes the moment of inertia around the x-axis when the objects are rotated around the x-axis, Ixy denotes the moment of inertia around the y-axis when the objects are rotated around the x-axis, and so on.

These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has

\mathbf{I}=\iiint_V  \rho(x,y,z)\left( \|\mathbf{r}\|^2 \mathbf{E}_{3} - \mathbf{r}\otimes \mathbf{r}\right)\, dx\,dy\,dz,

where \mathbf{r}\otimes \mathbf{r} is their outer product, E3 is the 3 × 3 identity matrix, and V is a region of space completely containing the object. Alternatively, the equation above can be represented in a component-based method. Recognizing that, in the above expression, the scalars Iij with i\ne j are called the products of inertia, a generalized form of the products of inertia can be given as

{{I}_{ij}}=\iiint_{V}{\rho \left( \mathbf{r}\centerdot \mathbf{r}{{\delta }_{ij}}-{{r}_{i}}{{r}_{j}} \right)\,dV}

The diagonal elements of I are called the principal moments of inertia.  

 Derivation of the tensor components

The distance r of a particle at \mathbf{x} from the axis of rotation passing through the origin in the \mathbf{\hat{n}} direction is  |\mathbf{x}-(\mathbf{x} \cdot \mathbf{\hat{n}}) \mathbf{\hat{n}}|. By using the formula I = mr2 (and some simple vector algebra) it can be seen that the moment of inertia of this particle (about the axis of rotation passing through the origin in the \mathbf{\hat{n}} direction) is   I=m(|\mathbf{x}|^2 (\mathbf{\hat{n}} \cdot \mathbf{\hat{n}})-(\mathbf{x} \cdot \mathbf{\hat{n}})^2) This is a quadratic form in \mathbf{\hat{n}} and, after a bit more algebra, this leads to a tensor formula for the moment of inertia

 {I} = m [n_1,n_2,n_3]\begin{bmatrix}  y^2+z^2 & -xy & -xz \\ -y x & x^2+z^2 & -yz \\ -zx & -zy & x^2+y^2 \end{bmatrix} \begin{bmatrix}  n_1 \\  n_2\\ n_3 \end{bmatrix} .

This is exactly the formula given below for the moment of inertia in the case of a single particle. For multiple particles we need only recall that the moment of inertia is additive in order to see that this formula is correct.

Reduction to scalar

For any axis \hat{\mathrm{n}}, represented as a column vector with elements ni, the scalar form I can be calculated from the tensor form I as

 I = \mathbf{\hat{n}^\top} \mathbf{I}\, \mathbf{\hat{n}} =  \sum_{j=1}^{3} \sum_{k=1}^{3} n_{j} I_{jk} n_{k}

The range of both summations correspond to the three Cartesian coordinates.

The following equivalent expression avoids the use of transposed vectors which are not supported in maths libraries because internally vectors and their transpose are stored as the same linear array,

 I = \mathbf{{I}^\top} \mathbf{\hat{n}} \cdot \mathbf{\hat{n}}

However it should be noted that although this equation is mathematically equivalent to the equation above for any matrix, inertia tensors are symmetrical. This means that it can be further simplified to:

 I = \mathbf{{I}} \mathbf{\hat{n}} \cdot \mathbf{\hat{n}}

 Principal axes of inertia

By the spectral theorem, since the moment of inertia tensor is real and symmetric, it is possible to find a Cartesian coordinate system in which it is diagonal, having the form

 \mathbf{I} = \begin{bmatrix} I_{1} & 0 & 0 \\ 0 & I_{2} & 0 \\ 0 & 0 & I_{3} \end{bmatrix}

where the coordinate axes are called the principal axes and the constants I1, I2 and I3 are called the principal moments of inertia. The principal axes of a body, therefore, are a cartesian coordinate system whose origin is located at the center of mass. The unit vectors along the principal axes are usually denoted as (e1, e2, e3). This result was first shown by J. J. Sylvester (1852), and is a form of Sylvester's law of inertia. The principal axis with the highest moment of inertia is sometimes called the figure axis or axis of figure.

When all principal moments of inertia are distinct, the principal axes are uniquely specified. If two principal moments are the same, the rigid body is called a symmetrical top and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a spherical top (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order m, i.e., is symmetrical under rotations of 360°/m about a given axis, the symmetry axis is a principal axis. When m > 2, the rigid body is a symmetrical top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, e.g., a cube or any other Platonic solid.

The motion of vehicles is often described about these axes with the rotations called yaw, pitch, and roll.

A practical example of this mathematical phenomenon is the routine automotive task of balancing a tire, which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.

 Parallel axis theorem

Once the moment of inertia tensor has been calculated for rotations about the center of mass of the rigid body, there is a useful labor-saving method to compute the tensor for rotations offset from the center of mass.

If the axis of rotation is displaced by a vector R from the center of mass, the new moment of inertia tensor equals

 \mathbf{I}^{\mathrm{displaced}} = \mathbf{I}^{\mathrm{center}} + m \left[ \left(\mathbf{R} \cdot \mathbf{R}\right) \mathbf{E}_{3} - \mathbf{R} \otimes \mathbf{R} \right]  


6 years ago

Post Your Answer

Other Related Questions on Mechanics

a packet is released from a balloon accelerating upward with acceleration a. the acceleration of packet just after the release is
Since the ballon was moving upward with acceleration a, there is a force of gravity on the packet too. The balloon is moving upward bexause the reaction force R>W, weight of the packet. As...
Shaswata Biswas one month ago
Just after the release the packet posses only the component of the velocity aquired. Here the accleration of the balloon has no effect on the acceration of the body. When the body is...
Shaswata Biswas one month ago
Shaswata Biswas your anwer is right. But can you please explain why the acceleration of the balloon has no effect on the acceleration of the body?
Dhanyashree one month ago
2kg of gas at a pressure of 1.5 bar.occupies a volume of 2.5m cube.if this gas compresses isothermally to 1/3 times the initial volume.find temperature,work done,heat transfer.
As the pocess is isothermal chanege in Internal energy would be zero and hence temperature remains costant that is T. From ideal gas eqn. PV =nRT 1.5 * 2.5*1000 = 2000/32 * 0.0821 * T we...
2017 years ago
Is there any book on theory of relative velocity of approach and separation
@ manju h c ver ma is the best one for that , it explains the relative velocity part in a little bit of detail by taking the example of diffrent scenario . so, u can use that , they also...
Umakant biswal 3 months ago
I think cengage physics mechanics-1 is the best because there is a complete explaination about all cases like minimum time of approach also angle of particles without using calculus and in...
benny bhai 3 months ago
DC Pandey explains the topic clearly andcan be very helpful to do problems related to relative velocity.So I would recommend you Mechanics-I by DC Pandey.
Mudit 3 months ago
Define scalar and vector quantities and distinguish between them
Scalar quantity basically associates with one dimensional quantity. But the vector quantity has direction with it. For example-> Force, Displacement, Area etc. Scalar includes current,...
Vikas TU 2 months ago
vector is a physical quantity which has both magnitude and direction.ex:velocity,acceleration,force,weight.
k sowmya 2 months ago
‘n' identical cells are joined in series with two cells A and B with reverse polarities .EMF of each cell is ‘E' and internal resistance 'r'. Calculate the potential difference across the...
netresistance = nr net voltage in series = ne E1 = e + ir for n cells and n resitances, E2 = ne + i(nr) potential diff. E1 – E2. hence.
Vikas TU 2 months ago
Tell me briefly about magnetism??? And Explain??????????????
magnetism is the phenomenon which have both attractive and repulsive power . elcetric monople exist where as magnetic monoploe doesn’t exist. electric force exist between the charges at...
vannala shivanand 10 months ago
dear chetan...Magnetism is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic...
mohan 9 months ago
dear chetan...Magnetism is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic...
mohan 9 months ago
View all Questions »

  • Complete Physics Course - Class 12
  • OFFERED PRICE: R 2,600
  • View Details
Get extra R 650 off

  • Complete Physics Course - Class 11
  • OFFERED PRICE: R 2,800
  • View Details

Get extra R 700 off

More Questions On Mechanics

Ask Experts

Have any Question? Ask Experts

Post Question

Answer ‘n’ Earn
Attractive Gift
To Win!!!
Click Here for details