Video Transcript
A new particle has been discovered
with a charge of 3.2 times 10 to the negative 19th coulombs and a charge-to-mass
ratio of 4.45 times 10 to the seventh coulombs per kilogram. What is the mass of the
particle? Give your answer to three
significant figures.
The question is asking us to find
the mass of a particle given its charge and its charge-to-mass ratio. This is possible because the
charge-to-mass ratio provides a relationship between an object’s charge and its
mass. Specifically, the charge-to-mass
ratio is a particle’s charge divided by its mass. If we plug in values from the
question, we have 4.45 times 10 to the seventh coulombs per kilogram is equal to 3.2
times 10 to the negative 19th coulombs divided by the unknown mass. To solve for mass, we multiply both
sides by mass and divide by 4.45 times 10 to the seventh coulombs per kilogram.
On the left-hand side, we have 4.45
times 10 to the seventh coulombs per kilogram in both the numerator and the
denominator, so the division leaves us with just mass. On the right-hand side, mass in the
numerator divided by mass in the denominator is just one, and we’re left with 3.2
times 10 to the negative 19th coulombs divided by 4.45 times 10 to the seventh
coulombs per kilogram. All that’s left is to
calculate. Let’s start off by dividing 3.2
times 10 to the negative 19th by 4.45 times 10 to the seventh. Plugging into a calculator gives us
7.19101 and several more decimal places times 10 to the negative 27th. As for the units, we have coulombs
divided by coulombs per kilogram. Coulombs in the numerator divided
by coulombs in the denominator is just one. And having per kilograms in the
denominator is equivalent to having just kilograms in the numerator. So the overall units of our answer
will be kilograms, which is good because we’re looking for a mass.
Now we just need to round our
answer so it has three significant figures. The first three significant digits
of our answer are seven, one, and nine. Looking at the next digit, one is
less than five, so nine doesn’t change when we round to three significant
figures. So to three significant figures,
the mass of our particle is 7.19 times 10 to the negative 27th kilograms. As it happens, this mass that we’ve
calculated and the provided charge match up very closely with the mass and charge of
an 𝛼 particle.
