The Physics Philes, lesson 71: A Weighty Situation

Last week we started our journey into gravity, undeniably the best of all the fundamental forces. We continue our journey today with an examination of the relationship between gravity and weight.

We talked about weight in a bit of a round-about way back in lesson 6 when I covered Newton’s second law of motion, or force equals mass times acceleration. You see, your weight is actually just the measurement of the force that is pulling your body down on a scale. Now that we’re talking about gravity, we can more fully understand what “weight” actually means.

The definition of weight, according to my trusty textbook, is “the total gravitational force exerted on a body by all other bodies in the universe.”

Whoa. That kind of makes stepping on a scale feel kind of grand, doesn’t it? This is the measurement of gravity from Earth and Saturn and the sun and Alpha Centauri A and B acting on little ol’ me! That’s very cool.

But even though the force of gravity form those far-flung places may technically be there, they are too far away to really matter. When we are actually on a place – like Earth or the moon or that secret parallel Earth on the other side of the sun – we can ignore all that other gravity and focus on the gravitational force exerted by that place.

OK, so weight is the gravitational force that acts on a body at a particular place. We know that the moon exerts a weaker gravitational force than the Earth, and this makes sense if you remember Newton’s universal law of gravitation that we discussed last week. We can use the law of gravitation to figure out the acceleration due to gravity. To do that, we need to use the second law of motion.

First, let me refresh your memory. We know the law of gravitation:

Screen shot 2013-10-26 at 8.03.59 PM

and we know the second law of motion, written in terms of weight (w) and acceleration due to gravity (g):

Screen shot 2013-10-26 at 8.05.35 PM

This is all we need to derive the equation to find the force of gravity anyplace in the universe! Don’t you feel powerful? Let’s do it!

Since we know that weight is just a force, we can replace F in the gravity equation with mg, like so:

Screen shot 2013-10-26 at 8.13.25 PM

Let’s have m_P be the mass of the planet/moon/Death Star in question. All we need to do is divide by m and solve for g:

Screen shot 2013-10-26 at 8.14.55 PM

Voila! The acceleration due to gravity, anywhere! And look, it doesn’t even matter what the mass of the second object is. Acceleration due to gravity is completely independent of that mass of that second body. Kind of neat, right? Guess what else we can do with the law of gravity. We can determine the mass of the Earth! And it’s not even hard!

All we have to do is solve for m in the above equation.

Screen shot 2013-10-26 at 8.22.14 PM

We know the radius of the Earth is 6. 38 x 10^6 m, the acceleration due to gravity on earth is 9.8 m/s^2, and the gravitational constant is 6.67 x 10^-11. We can just plug in those values and get the mass of the Earth!

Screen shot 2013-10-26 at 10.36.19 PM

Screen shot 2013-10-26 at 10.37.39 PM

This is really close to 5.974 x 10^24 kg – the accepted value for the mass of the Earth!

This equation doesn’t just work for someone’s weight on a planet. We can find how much someone weighs above a planet, as well. Let’s use the Earth because, well, it’s here.

Let’s say that you’re an astronaut floating above the Earth. Let R be the radius of the Earth and r be the distance from you to the center of the Earth. (That would make your distance above the surface of the Earth r – R.) All we have to do is replace the square of the radius of the planet with the square of your actual distance from the center of the Earth.

It’s important to remember that the weight we measure on Earth is only our apparent weight. It differs slightly from the Earth’s gravitational pull because the Earth rotates, which means it’s not a perfect inertial frame of reference. But we won’t mess with that too much right now.

That’s all for now. But don’t worry. There are lots of cool gravity things to come.

Featured image credit: Beth Scupham via Flickr

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Mindy is an attorney and Managing Editor of Teen Skepchick. She hates the law and loves stars. You can follow her on Twitter and on Google+.

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