Sponge - I love the boat ride at Six Flags. If the radius of the ride is 6 meters and the boat completes the circle in 10 seconds, what is the centripetal acceleration? What centripetal force is required on a child whose mass is 30 kg?

Circular Orbits - There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.

FC = G mME /r2 = mv2/r

v = √GME/r

v = √GME/r

Mass (m) cancels out of the equation; therefore, a satellite with a large mass has exactly the same orbital speed as a satellite with a small mass.

Ex. 9 - Determine the speed of the Hubble space telescope orbiting at a height of 596 km above the earth’s surface.

Ex. 10 - The characteristics of light detected in the galaxy M87 by the Hubble Space Telescope indicate that matter is orbiting at a speed of 7.5 x 105 m/s at a distance from the center of 5.7 x 1017 m. Find the mass M of the object at the center of this galaxy.

The period of a satellite is the amount of time required for one orbit.Remember: v = 2πr/ T, and

v = √ GME/r,

So....

So . . . 2πr/ T = √ GME/r

Solving for T:

T = 2πr3/2/√GME

The fact that the period is proportional to the three-halves power of the orbital radius is Kepler’s Third Law of Planetary Motion.

Synchronous satellites are put into a circular orbit that is in the plane of the equator. All synchronous orbits must orbit at the same height above the surface of the Earth, as the equation suggests.

Ex. 11 - What is the height above the Earth’s surface at which all synchronous satellites (regardless of mass) must be placed in orbit?

We say that people or objects on a satellite are “weightless”. They are actually apparently weightless because they and the satellite are in free-fall.

Rotational motion of a space station would produce apparent gravity due to the centripetal force produced.

Ex. 13 - At what speed must the surface of a cylindrical space station (r = 1700 m) move so that an astronaut inside will experience a push on his feet that equals his own weight?

Ex. 14 - A space laboratory is rotating to create artificial gravity. Its period of rotation is chosen so the outer ring (rO = 2150 m) simulates the acceleration of gravity on Earth (9.80 m/s2). What should be the radius rI of the inner ring, so it simulates the acceleration of gravity on the surface of Mars (3.72 m/s2)?

In vertical circular motion the centripetal force is the vector sum of the normal force and the component of the weight that pushes directly toward the center of the circle. In the lower half of the circular motion the centripetal force will be less (normal force minus weight) than in the upper half of the circle (normal force plus weight).

A the top of the circle, the centripetal force is normal force plus weight:

FC = mv2/r = FN + mgAt the correct speed, normal force can become zero and:

FC = mv2/r = mgSolving the last two terms for v gives: v= √rg

At this speed, the track does not exert a normal force; mg provides all the centripetal force. The rider at this point experiences apparent weightlessness.

Ex. 15 - A roller coaster loop has a radius of 10 meters. What is the minimum velocity required to keep the cars in the loop during the ride?

Ex. 16 - An evil father is pushing his daughter in a swing. If he gleefully pushes her as hard as he can, what is the minimum velocity at which she will make a complete vertical circle on the swingset if the swing’s chain is 6 meters long?

Ex. 17 - After a lengthy trial, the court decides that the punishment should fit the crime. The father is sentenced to be pushed in a circle on the same swingset. If he weighs six times as much as his daughter, what is his minimum speed to complete the circle? What other differences are there in the execution of his punishment?