Gravitation - Unit 7

Gravitation - Unit 7 The magnitude of gravitational force between 2 masses is determined by Newton’s Universal Law of Gravitation. Given 2 masses m1 and m2, and the distance r12 between them, the force that m1 would exert on m2 is: 𝐹12 = 𝐺 𝑚1 𝑚2 𝑟̂ |𝑟12 |2 12 If we are only worried about the magnitude of force, then we can use this form of the law. |𝐹𝐺 | = 𝐺𝑚1 𝑚2 𝑟2 G in these equations is the gravitational constant, which is about 6.674 × 10−11 Nm2/kg2. Non-Point Masses If you have objects that cannot be treated as point masses (like symmetrically spherical planets) and take up space (have spatial extent), you need to think a little differently to work with them. Firstly: when a mass M is subjected to forces from many other points, the net gravitational force is simply the vector sum of all of the forces that are acting on it. 𝑁 𝐹=∑ 𝑖 𝐹𝑖 = 𝐺𝑀 (∑ 𝑖=1 𝑚𝑖 𝑟𝑖2 𝑟̂𝑖 ) As 𝑁 ⟶ ∞, this summation becomes an integral that we can use to describe objects that take up space. 𝐹=∫ 𝑑𝐹 = 𝐺𝑀 ∫ 𝑑𝑚 𝑟̂ 𝑟2 Gravitational Field Generally we describe gravitational force as: 𝐹 = 𝑚𝑔 To find g, we need to group the variables from Newton’s Law of Universal Gravitation. 𝐹 = 𝑚 [− 𝐺𝑚1 𝑟̂ ] = 𝑚𝑔 |𝑟|2 g is a vector field function, and is also the acceleration due to gravity. Like the electric field E and magnetic field B in from E&M, g is also known as the gravitational field in field theory. When we are close to Earth’s surface and our distance from Earth’s center r doesn’t change very much, g can be treated as a constant. If 𝑚1 = 𝑚𝑒𝑎𝑟𝑡ℎ = 5.972 × 1024 𝑘𝑔 and 𝑟 = 𝑟𝑒𝑎𝑟𝑡ℎ = 6.371 × 106 𝑚, we can compute a value of 9.81 m/s2 for g. In general, the gravitational field g generated by a point mass m shows how it influences the gravitational forces on other masses. 𝑔(𝑚, 𝑟) = − 𝐺𝑚 𝑟̂ |𝑟|2 The net gravitational field at a position r is given by the vector sum of all the gravitational fields acting there, just like gravitational force. 𝑔=∑ 𝑖 𝑔𝑖 = 𝐺 (∑ 𝑖 𝑚𝑖 𝑟𝑖2 𝑟̂𝑖 ) We can also describe this with an integral for objects with spatial extent. 𝑔=∫ 𝑑𝑔 = 𝐺 ∫ 𝑑𝑚 𝑟̂ 𝑟2 Gravitational Potential Energy Gravity is a conservative force, and that means that masses in a gravitational field have potential energy. Gravitational potential energy of something in Earth’s gravitational influence is given by the equation: 𝑈𝑔 = − 𝐺𝑚𝑒𝑎𝑟𝑡ℎ 𝑚𝑜𝑏𝑗𝑒𝑐𝑡 𝑟 Obviously you can replace the mass of the Earth with some other body to examine other orbits. You can see that gravitational potential energy is always negative, unless you are an infinite distance away from Earth (or some other planet), where potential energy would be zero. Circular Orbit Velocity If we have a small mass m in a circular orbit around a larger mass M, then the centripetal force required for the circular motion is supplied by gravitational force. 𝐺𝑀𝑚 𝑚𝑣 2 𝐹𝑔 = 𝐹𝑐 ⟶ = 𝑟2 𝑟 We solve for v and get the orbital speed (or orbital velocity), which does NOT depend on the mass of the object that is in orbit. 𝑣𝑜𝑟𝑏𝑖𝑡 = √ 𝐺𝑀 𝑟 Note that this ONLY works for objects that are very small in comparison to what they orbit. Orbital Period: Kepler’s Third Law Kepler’s Third Law states that the square of the orbital period of something orbiting something else is directly proportional to the cube of the semi-major axis of its orbit (a circle is a special ellipse: the semi-major axis is just the radius). We can derive the orbital period of a circular orbit to see this law in action. 𝑇= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 4𝜋 2 𝑟 2 4𝜋 2 𝑟 3 2 𝑇= ⟶𝑇 = ⟶𝑇 = 𝐺𝑀 𝐺𝑀 𝐺𝑀 √ 𝑟 𝑟 2𝜋𝑟 2 Conservation of Angular Momentum and Mechanical Energy Gravity is always centripetal (pointing to the center), and is a conservative force. There are 2 important consequences of orbits that come from this. Firstly: ANGULAR MOMENTUM IS ALWAYS CONSERVED. Angular momentum is only affected by forces acting tangentially: centripetal forces never affect angular momentum. Angular momentum is therefore conserved in any type of orbit (even in elliptical orbits, where it will be important. Secondly: MECHANICAL ENERGY IS ALWAYS CONSERVED. Gravity is a conservative force, so the sum of the kinetic energy of the orbiting object and the potential energy in the system will always stay the same. These 2 properties of orbits mean that even if an orbit isn’t perfectly circular (like an elliptical orbit), you can determine things about the orbit based on velocity, distance from the central body, etc.