Newton’s Universal Law of Gravitation is introduced, and a new way to determine your weight is discussed.
Newton’s Universal Law of Gravitation is introduced, and a new way to determine your weight is discussed. (1:17) Changes to gravitational force are evaluated. (4:34) Episode 12 helps you learn the physics of space travel by looking at orbital velocity (5:20) and orbital period (7:15).
The Question of the Day asks: (9:37)
How can you quadruple the force of gravity between two objects?
a) quadruple one of the masses
b) double both masses
c) half the distance between the objects
d) all of the above
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on gravitational fields, orbital period, and orbital velocity.
Let’s Zoom out:
Unit 3 – Circular Motion and Gravitation
Topic 3.6-3.8
Big Idea – Force Interactions
Let’s Zoom out:
Floating in deep space, an astronaut releases a wrench while making repairs on his ship. What can he do? How will he retrieve it? The good news… he doesn’t have to do anything! He has mass, and the wrench has mass, so over time the gravitational field that they each exert on each other will allow for the astronaut and his wrench to accelerate towards each other veeeeeerrrrrryyyy sloooooowwllly. Thanks goodness for Newton’s less well known law, the Universal Law of Gravitation.
Let’s Zoom in:
The universal law of gravitation states that a gravitational force exists between two masses with the force directly proportional to the masses involved and inversely proportional to the square of the distance between the object’s centers of mass. Phew! Newton knew this, but what he didn’t know was that somehow there was also a very small constant involved as well. The gravitational constant or big “G” is equal to 6.67 x 10-11N*m2/kg2. These units are fairly easy to remember because when the constant is multiplied by the mass of each object, and divided by the distance squared, you need to have a force unit remaining. But… also because it is printed on the Physics 1 equation sheet.
That’s simple enough. If you wanted to find your own force of gravity on Earth, you would multiply your mass (maybe 50 kg) by the Earth’s mass (6 x 1024kg) and by the gravitational constant, then divide by the distance between your center of mass and the earth’s center of mass squared. So, basically the radius of the Earth (6.4 million meters) squared or 4.1 x 1013 m2 and we get a weight or force of gravity of about 490 N. Well, that isn’t news to us, we could have just multiplied your 50 kg mass by 9.8 m/s2 and arrived at the same weight. YES! They both work. Well, then that means that your mass exists in a gravitational field causing all objects to accelerate at the same rate as long as they are on Earth and at roughly the surface of the planet. In fact, you should be able to come up with the gravitational acceleration (little “g”) on the surface of any planet by multiplying the planet’s mass by the gravitational constant (big “G”) and dividing by the distance to the planet’s center squared. Try it out. No seriously, search for NASA planetary data and look up the mass and radius of the planet of your choosing. You will now be able to accurately determine the gravitational acceleration on that planet’s surface. Pretty cool huh?
What if you increase the distance between your center and the planet’s center? For example, maybe you are on a space mission and are located twice as far as before when you were on the planet’s surface? Well, if the distance is doubled, the universal law of gravitation indicates that the force is inversely related to the square of the distance. But, what does that mean? It means if you double the distance, then it gets squared as well, and since it is in the denominator, the outcome on the resulting gravitational force is a ¼ multiplier. You’d feel a fourth of the gravitational force. What if your mass was doubled? Well, mass is directly proportional to the gravitational force, so the force would also be doubled.
So what? I mean, I would love to visit another planet as I am sure some of you might, but knowing your weight on that planet is a bit of a factoid. What good is all of this? Well, space agencies around the world have learned much about other planets by sending satellites to collect data, and it turns out, it is pretty important to know about gravitational forces if you want to send a probe into orbit. Take Mars for example, numerous unmanned crafts are currently in orbit above Mars. What is an orbit? Well, an orbit is what happens when you travel fast enough in the forward direction that as you fall back to the planet due to gravity, you match the curvature of the planet itself. As a result you just keep falling, and falling, and falling. The speed which you need to achieve is called the orbital velocity. If you recognize that the gravitational force of the planet is a net force, as its causing you to move in a circular path, then you can see that you can set gravitational force equal to the centripetal force. So for Mars which has a mass of 6.4 x 1023 kg and a radius of 3.4 million meters, you can multiply the mass of Mars by the gravitational constant divided by the radius to solve for the velocity squared from the centripetal force side of the equation. Taking a square root of velocity gets you the orbital velocity of 3543 m/s or 7928 miles/hour. Wait, you mean you don’t need the mass of the spacecraft? Nope. It cancels since both equations have the mass of the object doing the orbit.
Not too shabby. Not tooooo shabby at all. Knowing how fast you’re going while traveling through space is very important as you can imagine, but so is how long you travel for so you can work out travel distances etc. Additionally, timing is essential when it comes to communicating with spacecraft via radio waves millions of miles across space. The time it takes a spacecraft to complete a single orbit is known as the orbital period. For our spacecraft traveling around Mars, you can imagine its path around the planet as making a circle which has a circumference of 2. The orbital period is represented with a capital “T” and if you know the speed and the distance then finding time is easy for an object with constant speed. Distance over time is average speed. So, for our craft, 2π times the radius of Mars (3.4 million meters) divided by the period is equal to the orbital speed we found of 3543 m/s around Mars. We get an orbital period of 6030. Paying attention to units, you will notice that our orbital period is in seconds. Not always ideal, so we can convert to any time unit we like. It takes just over an hour and a half to orbit Mars at a height right above the surface. If our craft was set a good deal above the surface, and most are to avoid atmospheric drag, then we would have to add that height to the planet’s radius to come up with the radius of the orbit made by the craft.
To Recap…
All pairs of masses apply a gravitational force on each other that is equal and opposite. The gravitational force is inversely proportional to the distance between two object’s centers of mass squared, and directly proportional to their masses. Orbital speed can be found and used by setting gravitational force equal to the centripetal force. Orbital period can be determined if you divide the circumference by the orbital speed as long as all of the measurement units are in agreement.
Coming up next on the APsolute RecAP Physics 1 Edition, we will look at the world in a completely new way by using the concepts of work and energy.
Today’s Question of the Day focuses on gravitation.
Question: How can you quadruple the force of gravity between two objects?
a) quadruple one of the masses
b) double both masses
c) half the distance between the objects
d) all of the above