Key Terminology
Angular Speed/Velocity - A change in angular position over time
Angular speed = change in angle/change in time
Tangential Speed/Velocity - Speed an object is moving tangent to the path
Tangential speed = angular speed (in radians/second) * radius
Angular speed = change in angle/change in time
Tangential Speed/Velocity - Speed an object is moving tangent to the path
Tangential speed = angular speed (in radians/second) * radius
Uniform Circular Motion
Uniform circular motion is the motion of an object travelling at a constant speed along a circular path.
From: https://qph.fs.quoracdn.net/main-qimg-e4c2a0627d87511bfc5690c9ef7edf9d
When an object spins with constant velocity, it actually has an acceleration because its direction changes. This acceleration is called centripetal acceleration. Centripetal means "center-seeking". Centripetal acceleration is an acceleration that points towards the center of rotation. From the circular motion lab, we proved the equation: centripetal acceleration = velocity^2/radius. If velocity doubles, centripetal acceleration would quadruple. If the radius doubles, centripetal acceleration would be halved. From Newton's Second Law, we also know that acceleration = net force/mass. Often times we can use the two equations to solve for unknown variables.
Universal Gravitation
Gravity is proportional to mass of each body and inversely proportional to the square of the distance between the two bodies. Henry Cavendish performed an experiment in 1798 where he accurately measured the gravitational force between two objects. Today we know that the force of gravity between two objects = the universal gravitational constant G (6.67*10^-11) times the mass of the first object times the mass of the second object divided by the distance between them squared. Fg = G(m1*m2)/r^2. If the mass of one object doubles, the gravitational force on each other also doubles. If the radius between the two objects doubles, the gravitational force quarters.
Cavendish's experiment, from OneNote
Gravitational Field
Field theory relies on an object creating a field which exists everywhere in space. Other objects interact with the field and not the object that generates it. The Earth's gravitational field, small g, can be calculated by g = Fg/m, m is the mass of an object on earth. From the universal gravitation equation above, we can find that Fg = G*m*mE/rE^2, where mE is the mass of the Earth and rE is the distance from Earth's center to its surface. Plugging this back into the equation, we get g = (G*m*mE)/rE^2/m = G*mE/rE^2. mE is about 5.972*10^24 kg and rE is about 6.38 * 10^6 m. We find that g equals the familiar value of 9.81 m/s^2.
Orbital Motion
Orbital motion occurs when an object is moving forward and at the same time is pulled by gravity toward another object. For example, the Moon is in orbital motion with the Earth. Orbital motion can be described by the equations of universal gravitation and centripetal acceleration. From Newton's Second Law, we know that net force = m*a. When an object is orbiting another object in space, the only net force acting on the satellite is the force of gravity from the bigger object. Thus Fg = m1*a, where m1 is the mass of the satellite. This is the same as Fg = m1*ac, where ac is the satellite's centripetal acceleration. Plugging in the equation for centripetal acceleration and universal acceleration, we get G*m1*m2/r^2 = m1*v^2/r, simplified to G*m2/r = v^2, where m2 is the mass of the bigger object (the planet). We can see that in this equation, circular motion does not depend on the mass of the satellite since m1 cancels out. Circular motion, however, does depend on the velocity the satellite travels at. If it travels at a high speed, it has to be close to the planet (a smaller radius). If the satellite travels at a slow speed, it has to be far away from the planet (a bigger radius). A good visualization of this concept is Newton's Cannon (https://physics.weber.edu/schroeder/software/NewtonsCannon.html).
Field theory relies on an object creating a field which exists everywhere in space. Other objects interact with the field and not the object that generates it. The Earth's gravitational field, small g, can be calculated by g = Fg/m, m is the mass of an object on earth. From the universal gravitation equation above, we can find that Fg = G*m*mE/rE^2, where mE is the mass of the Earth and rE is the distance from Earth's center to its surface. Plugging this back into the equation, we get g = (G*m*mE)/rE^2/m = G*mE/rE^2. mE is about 5.972*10^24 kg and rE is about 6.38 * 10^6 m. We find that g equals the familiar value of 9.81 m/s^2.
Orbital Motion
Orbital motion occurs when an object is moving forward and at the same time is pulled by gravity toward another object. For example, the Moon is in orbital motion with the Earth. Orbital motion can be described by the equations of universal gravitation and centripetal acceleration. From Newton's Second Law, we know that net force = m*a. When an object is orbiting another object in space, the only net force acting on the satellite is the force of gravity from the bigger object. Thus Fg = m1*a, where m1 is the mass of the satellite. This is the same as Fg = m1*ac, where ac is the satellite's centripetal acceleration. Plugging in the equation for centripetal acceleration and universal acceleration, we get G*m1*m2/r^2 = m1*v^2/r, simplified to G*m2/r = v^2, where m2 is the mass of the bigger object (the planet). We can see that in this equation, circular motion does not depend on the mass of the satellite since m1 cancels out. Circular motion, however, does depend on the velocity the satellite travels at. If it travels at a high speed, it has to be close to the planet (a smaller radius). If the satellite travels at a slow speed, it has to be far away from the planet (a bigger radius). A good visualization of this concept is Newton's Cannon (https://physics.weber.edu/schroeder/software/NewtonsCannon.html).
Newton's Cannon, from OneNote
In addition, the reason that an astronaut feels weightless in a space station is not because that it is far away from the Earth (only about 400km), but because it is fast and has centripetal acceleration. The space station constantly falls to the Earth (and so does the astronaut) but it is travelling so fast that Earth's curvature bends around so it never crashes into the Earth. Because the astronaut is constantly in free fall, there is no normal force compressing on them so they don't feel their weight.