Centripetal force increases with rotational speed and mass. However, as the radius of curvature increases, the rate of change of velocity decreases, so the centripetal force is inversely proportional to the radius.
'Centrifugal' also comes from Latin, with the verb 'fugere', meaning "to flee the centre". It is what is experienced by the passengers in a car when a car goes around a bend or in a circle. The car pushes against the passengers, whose inertia would otherwise cause them to continue moving in a straight line. The passengers feel a 'force', but it is really the result of the centripetal force causing them to change direction as they go around a circle. Since it is not really a force, we should more properly call it the 'centrifugal effect'.
Even though we do not think of it as such, a vehicle going around a bend in the road is actually undergoing circular motion that has a centre of curvature. The car's inertia causes the tyres to push against the road. That is why tyres need to be rough, to grip the road surface and create the friction which pushes the car at right angles to its motion.
Any object or person inside the vehicle will feel the centrifugal force - they 'sense' a push towards the outside of the car. It is not really a force. It is caused by their inertia, which wants to take them in a straight line. It is the vehicle which experiences the force of the road contact, not the passengers. Hence, it is the vehicle which moves under the passengers, not the passengers which move inside the vehicle, creating the illusion of a 'force'.
ac = v2/r, so therefore the force is: Fc = mv2/r
The centripetal force is equal to the centrifugal force on an object in circular motion. This force, Fc, is equal to the mass times the square of the tangential velocity, v, and inversely proportional to the radius, r.
A thought experiment is one done with the imagination. When it would be too difficult or impossible to do a real experiment, sometimes scientists think through an experiment with a logical argument to demonstrate some principle. Galileo made a famous thought experiment about gravity, which was eventually done by David Scott on the Moon. Albert Einstein proposed many thought experiments to test his Relativity theories, involving trains travelling close to the speed of light and boxes falling through space.
Isaac Newton also made a thought experiment about gravity and orbits: he imagined a very powerful cannon on top of a very high mountain. Then he asked the question: what would happen if the cannon fired a cannonball parallel to the ground? The answer was it would travel some distance while it fell to the ground. Then he asked what would happen if we make the cannon more and more powerful?
Gradually, the ball would travel further and further, until it was travelling around the curve of the planet's surface, 'missing' more and more of the planet. At a certain point, reasoned Newton, the cannonball would miss the entire planet, and since it still had the original velocity would pass the cannon and go around the planet again... and again. This was the first understanding of 'orbits'.
Just like Newton's cannonball, a satellite is continuously falling towards the Earth, but its horizontal velocity is so great it moves around the Earth. The Moon does the same, and the Earth, and all the planets, are in orbit around the Sun. The shape of the orbits is an ellipse. This is like a slightly stretched circle. But the principle of Newton's wonderful thought experiment still applies.
A satellite in orbit is in freefall, and is falling towards the Earth all the time. But is has a velocity that takes it at right angles to the fall. Therefore, it keeps moving to the side at just the right speed that it just misses all the time (it helps that the Earth is round - going around sharp corners would be a bit trickier).
Therefore, any object at the same altitude would fall at the same rate, and therefore have the same period of orbit.
The ISS International Space Station orbits the Earth every 92 minutes, at an altitude of about 400-420 km (remember, orbits are elliptical, so their altitude must vary as they go around).
Satellites in geostationary orbit are always over the same point of the Earth's surface (very useful for communications satellites).
Geostationary satellites are synchronised with the rotation of the Earth. Therefore, their orbit is the length of an Earth day, 24 hours.
Since they take a lot longer to go around the Earth than the ISS, their orbit is much higher. In fact, the geosynchronous orbit is at an altitude of 35,564 km (their orbit has a radius of 42,164 km, and the Earth's radius is 6,400 km).
Newton's Second Law of Motion says that a projectile will accelerate when in freefall. It also says that a projectile with horizontal velocity would experience exactly the same vertical acceleration as an object in freefall. The result is a parabola.
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