Always Check Your Units!

No matter what, units always have to add up. The equations must be homogeneous in terms of units. Checking your units can help you make sure that you haven't made any errors in manipulating equations, and sometimes even gives very quick solutions to hard problems.

Problem (F=MA 2014, Paraphrased): A paper helicopter with rotor radius $r$ and weight $W$ is dropped from a height $h$ in air with a density of $\rho$. The time it takes for some process to occur is

$$T=kh^\alpha r^\beta\rho^\gamma W^\omega$$

where $k$ is a dimensionless constant. Determine $\alpha$.

(a) $-1$ (b) $-1/2$ (c) $0$ (d) $1/2$ (e) $1$

Determine $\beta$.

(a) $1/3$ (b) $1/2$ (c) $2/3$ (d) $1$ (e) cannot be de termined

Solution: For part (a), we know that $T$ is in $s$, $k$ has no units, $h$ is in $m$, $r$ is in $m$, $\rho$ is in $kg/m^3$, and $W$ is in $N=kg\cdot m/s^2$. Then that means

$$s=(m)^\alpha(m)^\beta(kg/m^3)^\gamma(kg\cdot m/s^2)^\omega=m^{\alpha+\beta-3\gamma+\omega}\cdot kg^{\gamma+\omega}\cdot s^{-2\omega}$$

This gives the system $\alpha+\beta-3\gamma+\omega=0$, $\gamma+\omega=0$, and $-2\omega=1$ because of the cancellations that must occur. Therefore, $\omega=-\frac12$ and $\gamma=\frac12$, so $\alpha+\beta=2$.

But that doesn't solve for $\alpha$ or $\beta$! But, on the test, notice that the two questions are related. Since it's a multiple choice test, we can use process of elimination. Since we can find $\alpha$, we can find $\beta$, so (e) is wrong for #2. Therefore the only possible combination is $(\alpha,\beta)=(1,1)$!

We live in the world of numbers. There are hundreds of numbers connected with each and everybody. $2, 60, 20$ are several numbers connected with my friend, $Reham$. But these numbers actually don't make any sense and no one cant explain how these numbers are connected with Reham. After plugging the units after these numbers, they will appear with their significance in Reham's life. So, let's do it.

$2m$, $60kg$, $20 years$.

So, $2$ $m$ is the height of $Reham$. $60$ $kg$ is his weight and $20$ $years$ is his age. Like the above example, numbers have no value in physics until we plug units after them. But most of the time, we have a tendency to ignore the units during calculation which is very indecent and have a very large probability to make a wrong conclusion.

Why do we need to be careful about units during solving problems?

At the time of solving four of five steps problems, we need to play with a decent number of equations. These equations and steps can make your calculation four times harder if you don't pay attention on units.

So, let's give an example.

Suppose, I have thrown a ball vertically with some non-zero initial velocity $u=10m/s$. It takes $t$ seconds to reach at the maximum height. What is the value of maximum height?

Ans: It's a two step problem. I will solve it by using two different methods.

1st method:

Initial velocity is $u$.

At the maximum height, the final velocity $v$ will be $0$.

so, $$t=\frac{\left(v-u\right)}{g}$$

Now we know the value of t, because we have set it in an equation with all known values.

$h=\frac{1}{2}gt^2=\frac{1}{2}g\left(\frac{u}{g}\right)^2=\frac{1}{2}\frac{\left(u\right)^2}{g}$.

Now we can easily find out the value of height.

2nd method:

$t=\frac{-\frac{10m}{s}}{-\frac{10m}{s^2}}=1s$

so, $h=\frac{1}{2}gt^2=\frac{1}{2}\left(\frac{10m}{s^2}\right)\left(1s\right)^2=5m$

Between these above two methods which one will you recommend for calculation? First one, right?

Because using the second method can make your calculation messy and wrong. So, we had better choose the first one where we have less chance to make mistakes with units. Making mistake with units cant make any sense for your solution. If I would claim that my friend Reham is $60m$ tall, it would be a nonsense thing and improper statement about him. So, always check your units. We cant come down to a right conclusion by making mistakes with units. This is why the expert physics teachers always check the unit and the dimension of the solution of students and the good problem solvers always choose a procedure which makes less chance to make mistakes with units.