This is an old revision of the document!

# Chapter 10 - Systems of Particles and Conservation of Momentum

## Momentum

We can define momentum as the mass times the velocity

$\vec{p}=m\vec{v}$

Momentum has units $\mathrm{kg\,ms^{-1}}$

Unlike kinetic energy momentum is a vector quantity.

The most fundamental aspect of momentum is it's connection to force. We can express Newton's second law as

$\Sigma \vec{F} = \frac{d\vec{p}}{dt}=m\frac{d\vec{v}}{dt}=m\vec{a}$

Over a change in momentum $\Delta p$ from $mv_{1}$ to $mv_{2}$ in a time interval $\Delta t$ during which the instanteous force may vary the average force can also be a useful quantity to consider

$F_{ave}=\frac{\Delta p}{\Delta t}=\frac{mv_{2}-mv_{1}}{\Delta t}$

## Center of mass

We can consider the motion of any system of particles or extended mass to be composed of different kinds of motion with respect to the center of mass.

In general the displacement vector for the center of mass of a system of particles $m_{i}$ can be written as

$\vec{r}_{CM}=\frac{\Sigma_{i}m_{i}\vec{r_{i}}}{\Sigma_{i}m_{i}}=\frac{\Sigma_{i}m_{i}\vec{r_{i}}}{M}$

Differentiating with respect to time gives

$M\frac{d\vec{r}_{CM}}{dt}=\Sigma_{i}m_{i}\frac{d\vec{r}}{dt}$ or $M\vec{v_{CM}}=\Sigma_{i}m_{i}\vec{v}_{i}$

and doing it once more

$M\vec{a_{CM}}=\Sigma_{i}m_{i}\vec{a}_{i}$

$\Sigma_{i}m_{i}\vec{a}_{i}=\Sigma_{i}F_{i}=\Sigma \vec{F}_{ext}$

so we have obtained a new form of Newton's Second Law that works for a system of particles.

$M\vec{a_{CM}}=\Sigma \vec{F}_{ext}$

the total momentum of a system can also be written in these terms

$\vec{P}=M\vec{v}_{CM}$

## Center of mass for extended objects

For point like objects at given displacement the center of mass vector is

$\vec{r}_{CM}=\frac{\Sigma_{i}m_{i}\vec{r_{i}}}{M}$

For solid objects it is more useful to consider the center of mass in integral form.

$\vec{r}_{CM}=\frac{1}{M}\int \vec{r}\,dm$

which we should recall is the same as having a set of equations for each of the components, ie.

$x_{CM}=\frac{1}{M}\int x\,dm$ , $y_{CM}=\frac{1}{M}\int y\,dm$ , $z_{CM}=\frac{1}{M}\int z\,dm$

## Mass Density

For object's of uniform density we can express the mass element over which we integrate as a spatial element which in 3 dimensions is $dm=\rho\, dV$, in 2 dimensions is $dm=\rho_{A}\, dA$ in one dimension is $dm=\lambda\,dx$, where $\rho$, $\rho_{A}$ and $\lambda$ are the density, areal density or linear density of the object we consider.

In cases where we know the total mass and total size (either volume, area or length) of an object the density can be found by dividing the total mass by the total size.

## A uniform rod

We can consider a uniform rod to be a one dimensional object.

If we want to find the COM we could place the origin of our coordinate system at the center of the rod and then

$\vec{r}_{CM}=\frac{1}{M}\int \vec{r}\,dm=\frac{1}{M}\int_{-l/2}^{l/2}\lambda x\,dx=\frac{1}{M}\frac{\lambda}{2}((\frac{l}{2})^2-(-\frac{l}{2})^2)=0$

On the other hand if we were to place the origin at the left end of the rod then we could show that

$\vec{r}_{CM}=\frac{1}{M}\int \vec{r}\,dm=\frac{1}{M}\int_{0}^{l}\lambda x\,dx=\frac{1}{M}\frac{\lambda}{2}l^2$

and as $M=\lambda l$

$\vec{r}_{CM}=\frac{l}{2}\hat{i}$

## A thin uniform plate

We now look at a 2 dimensional object, in this case a thin rectangular plate.

The mass interval $dm=\rho_{A}\,dA=\rho_{A}\,dx\,dy$

The COM of the plate in the x direction is

$\vec{x}_{CM}=\frac{1}{M}\int x\,dm=\frac{1}{M}\int_{-l/2}^{l/2}\int_{-w/2}^{w/2}\rho_{A}x\,dy\,dx=0$

$\vec{y}_{CM}=\frac{1}{M}\int y\,dm=\frac{1}{M}\int_{-l/2}^{l/2}\int_{-w/2}^{w/2}\rho_{A}y\,dy\,dx=0$

We can see that symmetry often enables us to identify the center of mass of an object, equally distributed mass on either side of the origin cancels out.

If we instead put the bottom left hand corner of the plate at the origin then

$\vec{x}_{CM}=\frac{1}{M}\int x\,dm=\frac{1}{M}\int_{0}^{l}\int_{0}^{w}\rho_{A}x\,dy\,dx=\frac{1}{M}\frac{1}{2}\rho_{A}l^{2}w=\frac{l}{2}\hat{i}$

$\vec{y}_{CM}=\frac{1}{M}\int y\,dm=\frac{1}{M}\int_{0}^{l}\int_{0}^{w}\rho_{A}y\,dy\,dx=\frac{1}{M}\frac{w}{2}\rho_{A}l^{2}l=\frac{w}{2}\hat{j}$

## Center of mass of the human body

How symmetric do you think the human body is? Here's an experiment to determine the center of mass of people.

## Polar coordinates

For circular objects and rotational motion we will find polar coordinates to be advantageous.

To transform from polar coordinates in to Cartesian coordinates

$x=r\cos\theta$

$y=r\sin\theta$

and from Cartesian to polar

$r=\sqrt{x^{2}+y^{2}}$

$\large\tan\theta=\frac{y}{x}$

## COM of a thin uniform disk

$\vec{r}_{CM}=\frac{1}{M}\int \vec{r}\,dm=\frac{1}{M}\int\vec{r}\rho_{A}\,dA$

$dm=\rho_{A}\,dA=\rho_{A}r\,dr\,d\theta$

$x=r\cos\theta$

$y=r\sin\theta$

$x_{CM}=\frac{1}{M}\int x\rho_{A} \, dA=\frac{\rho_{A}}{M}\int_0^{2\pi}\int_{0}^{R}r^2\cos\theta\,dr\,d\theta=\frac{\rho_{A} R^3}{3M}\int_0^{2\pi}\cos\theta\,d\theta$ $=\frac{\rho_{A} R^3}{3M}[\sin(2\pi)-\sin(0)]=0$

$y_{CM}=\frac{1}{M}\int y\rho_{A} \, dA=\frac{\rho_{A}}{M}\int_0^{2\pi}\int_{0}^{R} r^2\sin\theta\,dr\,d\theta=\frac{\rho_{A} R^3}{3M}\int_0^{2\pi}\sin\theta\,d\theta$ $=\frac{\rho_{A} R^3}{3M}[-\cos(2\pi)+\cos(0)]=0$

## COM of a thin uniform half-disk

$\vec{r}_{CM}=\frac{1}{M}\int \vec{r}\,dm=\frac{1}{M}\int\vec{r}\rho_{A}\,dA$

$x_{CM}=\frac{1}{M}\int x\rho_{A} \, dA=\frac{\rho_{A}}{M}\int_0^{\pi}\int_{0}^{R}r^2\cos\theta\,dr\,d\theta=\frac{\rho_{A} R^3}{3M}\int_0^{\pi}\cos\theta\,d\theta$ $=\frac{\rho_{A} R^3}{3M}[\sin(\pi)-\sin(0)]=0$

$y_{CM}=\frac{1}{M}\int y\rho_{A} \, dA=\frac{\rho_{A}}{M}\int_0^{\pi}\int_{0}^{R}r^2\sin\theta\,dr\,d\theta=\frac{\rho_{A} R^3}{3M}\int_0^{\pi}\sin\theta\,d\theta$ $=\frac{\rho_{A}R^3}{3M}[-\cos(\pi)+\cos(0)]=\frac{2\rho_{A}R^3}{3M}$

We can convert this in to something more useful by considering that $M=\rho_{A}\frac{1}{2}\pi R^{2}$

which tells us that

$y_{CM}=\frac{4R}{3\pi}$

## Conservation of Momentum

In a collision in which no external force acts the total momentum of the system is a conserved quantity. This can be seen to be a consequence of Newton's third law.

Consider a collision between two objects

$m_{A}\vec{v}_{A}+m_{B}\vec{v}_{B}=m_{A}\vec{v'}_{A}+m_{B}\vec{v'}_{B}$

The force exerted by object A on object B must be

$\vec{F}_{AB}=\frac{d\vec{p}_{B}}{dt}$

And the force exerted by object B on object A must be

$\vec{F}_{BA}=\frac{d\vec{p}_{A}}{dt}$

But we know from Newton's Third Law that

$\vec{F}_{AB}=-\vec{F}_{BA}$

so that

$\frac{d\vec{p}_{A}}{dt}+\frac{d\vec{p}_{B}}{dt}=0$

## Conservation of momentum for many objects

More generally, the total momentum of a system

$\vec{P}=\Sigma_{i} p_{i}=\Sigma_{i} m_{i}v_{i}$

The change of this quantity with time

$\frac{d\vec{P}}{dt}=\Sigma_{i}\frac{d\vec{p_{i}}}{dt}=\Sigma_{i}F_{i}$

$F_{i}$ represents the net force on the $i$th body. All internal force have equal and opposite reaction force on some other body within the system (Newton's 3rd Law) and cancel out in the sum.

So

$\frac{d\vec{P}}{dt}=\Sigma_{i}F_{ext}$

In the absence of an external force the total momentum of a system is constant.

## Rockets

The key to rocket propulsion is that the momentum of the expelled gas is equal and opposite to the forward momentum of the rocket. This does not rely on any interaction with the external atmosphere.