Universal Law of Gravitation and Orbits AP Physics C

Universal Law of Gravitation

Newton’s Theory of Gravity Legend has it that Newton saw an apple fall from a tree, and it occurred to him that the apple was attracted to the center of the earth. If the apple was so attracted, why not the moon? Newton posited that gravity is a universal attractive force between all objects in the universe. Unfortunately he had no clue as to why. That’s what happens when you stare at the sun, poke your eyes with needles, and huff lead.

Universal Law of Gravitation Newton’s universal law of gravitation is as follows: 𝐹 𝑔 =−𝐺 𝑚 1 𝑚 2 𝑟 2 G is the gravitational constant. This is a universal constant with the value G = 6.67 x 10-11 N m2/kg2. This is not the same thing as g! The negative sign implies that the force is attractive. This means that all objects with mass attract one another with a gravitational force. The distance that we use is the distance from center of mass to center of mass.

Inverse-Square The force of gravity is an inverse square relation, meaning that as the separation distance increases, the force drops substantially. We often are simply concerned with the magnitude of the force of gravity, so often the formula is written this way: 𝐹 =𝐺 𝑚 1 𝑚 2 𝑟 2 When bringing up gravitational potential energy we will consider the force to be negative for constancy.

But that marker isn’t attracted to me Yes it is. Since G is so small, it means that the attractive force between two 1.0 kg masses, whose centers are 1.0 m apart, is 6.7 x 10-11 N. This is 100 billion times weaker than the force of gravity from the earth on either of the masses! Although weak, gravity is a long-range force. Gravity keeps the earth orbiting the sun and the solar system orbiting the center of the Milky Way galaxy.

Example Three stars are aligned in a row. The net force on the star of mass 2M is in what direction?

Gravitational Fields 𝑔= 𝐺𝑀 𝑟 2 Gravity is a field force, meaning that it is long range and can be represented by field lines. Little “g” is a measure of that strength of the gravitational field. 𝑔= 𝐺𝑀 𝑟 2

Falling through the Earth Suppose we dig a hole through the center of the Earth. If we jump into the hole, we experience an acceleration of 9.8 m/s2 at the surface. As we fall, the acceleration (and force) decreases. Once we reach the center of the planet, the net force is zero, but we continue moving due to our kinetic energy and momentum. After we pass the center, we experience a growing force towards the center, slowing us down. This means that we will oscillate!

Falling through the Earth The period of oscillation would be the exact same as a satellite orbiting at the surface of the Earth. If we approximate that the density of the Earth is constant (which it isn’t) and that you would not incinerate as you pass through the middle (which you would), we find that it would take approximately 84 minutes for one full oscillation. This also requires the Earth to be stationary (which it isn’t). Regardless, it would work for a stationary spherically symmetric rock the size of the Earth.

Gravitational Energy

Gravitational Potential Energy For objects near the surface of the Earth, we have used an approximation to describe the gravitational potential energy. ∆𝑈=𝑚𝑔∆𝑦 This is however, just an approximation. To really understand how gravitational potential energy works, we need to examine the universal law of gravity. Recall that a conservative force follows the following rule: 𝐹 𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒 = −𝑑𝑈 𝑑𝑥

Gravitational Potential Energy We now combine this rule with our formula for the force of gravity. 𝑑𝑈=−𝐹𝑑𝑟 𝑈 𝑟 =− ∞ 𝑟 − 𝐺𝑚𝑀 𝑟 2 𝑑𝑟 𝑈 𝑟 =𝐺𝑚𝑀 ∞ 𝑟 1 𝑟 2 𝑑𝑟 𝑈 𝑟 =𝐺𝑚𝑀∗ −1 𝑟 + 1 ∞ →𝑈 𝑟 = −𝐺𝑚𝑀 𝑟 This result may seem a little bit confusing at first glance. We choose infinity as our initial point because it is an easy reference point (a point from which to measure something). At infinity, we can say that the potential energy is equal to zero, because at that distance there would be no tendency for the objects to attract.

Gravitational Potential Energy Suppose two masses a distance r1 apart are released from rest. How will the small mass move as r decreases from r1 to r2? At r1 U is negative. At r2 |U| is larger and U is still negative, meaning that U has decreased. As the system loses potential energy, it gains kinetic energy while conserving mechanical energy. The smaller mass speeds up as it falls. Remember, only changes in potential energy have physical meaning.

Escape Velocity Consider a rocket leaving the earth. It usually goes up, slows down, and then returns to earth. There exists an initial minimum speed that when reached the rockets will continue on forever. Let's use conservation of energy to analyze this situation! 𝐾𝐸 𝑜 + 𝑈 𝑜 =𝐾𝐸+𝑈 𝐾𝐸 𝑜 = 1 2 𝑚 𝑣 𝑜 2 𝑈 𝑜 = −𝐺𝑚𝑀 𝑅 In this situation, we will say that the rocket can only truly “escape” the Earth if it gets to a distance of infinity. In this case, it would have a final velocity of zero.

Escape Velocity This means that the total mechanical energy of the system is equal to zero. Remember that defining the potential as negative does not carry any significant meaning. 𝐾𝐸 𝑜 + 𝑈 𝑜 =𝐾𝐸+𝑈 𝐾𝐸 𝑜 = 1 2 𝑚 𝑣 𝑜 2 𝑈 𝑜 = −𝐺𝑚𝑀 𝑅 1 2 𝑚 𝑣 𝑜 2 − 𝐺𝑚𝑀 𝑅 =0+0→ 𝑣 𝑜 = 2𝐺𝑀 𝑅 This is the minimum velocity to escape the gravitational pull of a planet! The M and R in the equation represent the mass and radius of the planet in question. Keep in mind that this only accounts for the object being launched and the planet it is launched from. When considering other planets and stars it becomes more complicated!

Kepler’s Laws and Orbits

Kepler’s Laws There are three laws that Johannes Kepler formulated when he was studying the heavens. 1st Law – The Law of Orbits: Planets move in elliptical orbits, with the sun at one focus of the ellipse. 2nd Law – The Law of Areas: A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. How is it that we know this? Angular momentum is conserved and thus constant! While further away from the sun, the planet can move slower, but when closer to the sun it has to move faster. 3rd Law – The Law of Periods: The square of a planet’s orbital period is proportional to the cube of the semi major-axis length. 𝑇 2 ∝ 𝑟 3

Linearizing Kepler’s Third Law You may have to use linearization in conjunction with Kepler’s third law. Suppose you are given a set of data that includes the distance of several orbiting bodies as well as their orbital period. If you graph R3 vs. T2, the slope can help you find the mass of the object that the satellites are orbiting. 𝑇 2 = 4 𝜋 2 𝐺𝑀 𝑟 3 →𝑠𝑙𝑜𝑝𝑒= 4 𝜋 2 𝐺𝑀 𝑀= 4 𝜋 2 (𝐺)(𝑠𝑙𝑜𝑝𝑒)

Orbital Speed Most orbits are elliptical, but we will focus almost exclusively on circular orbits, because circles are easy. In order to orbit in a perfect circle, the orbiting body must have a very particular speed. Any faster and the orbit becomes elongated and turns elliptical, any slower and the satellite comes crashing down to the planet. The orbital speed for circular orbit is independent of the mass of the satellite.

Example Two satellites have circular orbits with the same radius. Which has a higher speed? Two identical satellites have different circular orbits. Which has a higher speed?

Orbital Kinetic Energy We can substitute our orbital velocity equation in for the velocity term in kinetic energy to get a new expression for the kinetic energy of an orbiting body. 𝑣 2 =𝐺 𝑀 𝑟 , 𝐾𝐸= 1 2 𝑚 𝑣 2 𝐾𝐸= 𝐺𝑚𝑀 2𝑟 The question is WHY? Why do we need a new equation for kinetic energy? Well, the answer is that greatly simplifies the math. If we use regular kinetic energy along with potential, we will need both the orbital velocity AND the orbital radius. In this case, we need only the orbital radius.

Orbital Energy We can now create an expression for total mechanical energy of an orbiting object in terms of only the masses of the objects and the distance between them (center of mass to center of mass). 𝐸 𝑡𝑜𝑡𝑎𝑙 =𝐾𝐸+𝑈 𝐾𝐸= 𝐺𝑚𝑀 2𝑟 , 𝑈= −𝐺𝑚𝑀 𝑟 𝐸 𝑡𝑜𝑡𝑎𝑙 = 𝐺𝑚𝑀 2𝑟 + −𝐺𝑚𝑀 𝑟 = −𝐺𝑚𝑀 2𝑟 So by inspection we see that the kinetic energy function is always positive, the potential is negative and the total energy function is negative. In fact the total energy equation is the negative version of the kinetic. The negative is symbolic because it means that the mass “m” is BOUND to the mass of “M” and can never escape from it. It is called a BINDING ENERGY. Notice that we also have a relationship between orbital kinetic and potential energy: 𝐾𝐸=− 1 2 𝑈

Orbital Energetics The figure shows the kinetic, potential, and total energy of a satellite in a circular orbit. Notice how, for a circular orbit, total energy = U/2. It requires positive energy in order to lift a satellite into a higher orbit.

Orbital Energetics The figure shows the steps involved to lift a satellite to a higher circular orbit. The first kick increases KE without increasing U, so KE is not -U/2, and the orbit is elliptical. The satellite then slows down as r increases. The second kick increases KE again so that KE = -U/2, and the orbit is circular.