# Difference between Mass and Inertia

My questions is about mass and Inertia, what is the difference between Mass and Inertia? Are they the same or different? How?

I am really confused with it,some says mass is the measure of Inertia, if so the unit of Inertia is Kg?

Mass is one type of inertia.

Inertia is a general term for an object's resistance against acceleration (or against change in its velocity).

• In linear (translational) cases, the inertia is called mass $m$. The unit is $\mathrm{[kg]}$. The larger the mass, the tougher it is to push something to move or to slow something down.
• In rotational cases, the inertia is called moment of inertia $I=\sum mr^2$. The unit is $\mathrm{[kg\cdot m^2]}$. The larger the $I$, the tougher it is to swing a wheel up to fast spinning or to slow down the spinning.

The definition of inertia in both cases arrive from Newton's 2nd law (and it's equivalent rotational version):

$$\sum \vec F=m\vec a$$ $$\sum \vec \tau=I\vec \alpha$$

• I'm used to the notation $[m]=1\text{ kg}$, square brackets denote the unit of a quantity. – Jasper Aug 31 '18 at 13:35
• @Jasper Aha, I have not seen a notation like that before. Instead, I have often seen square brackets used to enclose units when shown separately without being tied to a value. Then a writing like $\mathrm{[N]=\left[kg\cdot \frac{m}{s^2}\right]}$ is "allowed" and used to focus on the unit correlation. – Steeven Aug 31 '18 at 14:03
• physics.stackexchange.com/questions/77690/… seems to exactly cover this issue. Seems to be a matter of taste. – Jasper Aug 31 '18 at 17:02

Newton's 1st law defines an inertial or Newtonian ,frame of reference , system . The quantitative measure of inertia within that system is called , mass . Suppose 2 objects, connected by a spring ,interact .How is the relative measure of inertia between objects A and B measured ?. Given that the objects are in the same frame of reference Their accelerations are In opposite directions and In a constant ratio . So dv[A]/dt = - k dv[B]/dt where k = m[B]/m[A] = relative measure of their inertias and is independent of units From here we find that m[A] dv[A]/dt = - m[B]/dv[B]/dt , this is the expression of Newton's 2nd law in so far as Change of motion is proportional to the force F = ma =m[A].d v[A]/dt in the direction of the motion. And since F[A] = - f[B], this embodies Newton's 3rd law.

So much for inertia . What might be confusing are the terms 'Moments of Inertia' and how are they used. As the name implies the are many moments ,1st moments and 2nd moments [ see statistics and use of moments to find the mean , the standard deviation and other properties]

1st MOMENTS are used to find CENTRES OF MASS. X = sigma[m(i).y(i)]/M , and Y = sigma[m(i).x(i)]/M , watch the positions of x(i) and y(i) , the levers , are linear , that is raised to the power of one .[x^1 ,y^1], tHEN [x,y] is the centre of mass

The 2nd moments are called MOMENTS OF INERTIA , is given the symbol I and is used to find kinetic energies Here the levers or the distances are squared and I = sigma[m(i).r(i)^2] =MR^2 To calculate the KE of a rotating body given the moment of inertia I,

KE =T = [1/2] MV^2 , BUT V= wR , V^2 =w^2R^2 So T = [1/2] [m.R^2].w^2 Then T =[1/2].Iw^2

As can be seen there is a link between the moment of inertia I and mass m Also between angular velocity w and linear velocity in the two equations

T = [1/2].mv^2 and T = [1/2] .Iw^2

See, inertia is just a (1)property of every body because of which the body stays in the same state in which it is. Like if its in motion it will remain in motion and if its in state of rest it will remain in state of rest until a force is applied on it. (2)It is also a property because of which the body tries to resist change in its state.

Now you asked mass is the measure of inertia: Note, the second point, it say the body tries to resist change in its state, actually the resistance is due to mass of the body.

Thus, if the mass is more inertia is more and vice-versa.