Physics12: Sheena Kuo Circular Motion & Gravity Centripetal Acceleration Centripetal Force Kepler's Laws Gravity on Earth Gravitational Potential Energy Escape Velocity - results from the change in the direction of velocity

- ; a is centripetal acceleration, v is

velocity, and R is radius of the curve; the unit is m/s^2

-The direction of a is along the velocity which is towards the center of a circle. The net force is constant, but it always changes direction. Thus the net force is perpendicular to the motion.

- Centripetal acceleration is resulted by... - is a net force toward centre, perpendicular to the motion Kepler's laws describe orbital motion and connect physics and astronomy. On Earth, the gravitational force is a specific constant. Issac Newton summarized this constant as 9.8 m/s^2. The expression of potential energy is Ep=mgh which g is valid as 9.8m/s^2.

h depends on the distance from the centre of gravity.

As a body is lifted, the work is done. Linear Acceleration - rate of velocity in time or how fast velocity

changes in a linear direction Satellite (artificial) - an object which is placed in orbit to observe astronomy

- there are different functions of a satellite, including mechanical fixing, connecting moblie, and scientific research.

- According to altitudes, satellites are classified as Low(0~2000km), Medium(2000~35,786km), and High(35,786km~) Earth Orbit. Apparent Weightlessness - Appears when you are in a structure and there is no detectable force against the body.

-The body is not exerting any force on the structure.

- Apparent weightlessness always appears in free fall!

- Apparent weightlessness appears on every satillite. Gravity still acts on the object, g, but the object accelerates -g. True Weightlessness - When a body is between two attraction bodies, the gravitational forces between the body and the planets are same and pull against each other.

So the body reaches weightlessness. Actual Weightlessness - a body moves away from Earth or other planet, the gravitational force decreases.

- The large distance, the weight approach to zero Universal Gravitation - a car stops at the red light and speeds up when the green light is on. The car accelerates to 60km/h after 5 seconds on a straight road. The change of speed here is 60km/h= 16.67 m/s, and the time interval is 5s. So the linear acceleration of this car is 3.33m/s^2. a c A 60-kg boy walks on a curve

with a speed of 3m/s. The curve

is actually a part of circle which has

a radius of 20m. Therefore, the

centripetal acceleration is

0.45m/s^2. V= 3m/s Acceleration should be

perpendicular to the motion Net Force The sum of all forces which apply on

an object. Those forces include frictional, tension, spring, normal, and applied forces F normal F gravity F applied F friction 1st Law 2nd Law 3rd Law The orbit of every planet is an ellipse with the Sun at one of the two foci. A line joining a planet and the Sun sweeps

out equal areas during equal intervals of time Every planet in Solar system: Mercury, Earth, Mars, Jupiter...etc. all surround the Sun in an ellipse shape. The areas that Earth sweeps for 10 days in a spring and in a fall are the same. The square of the orbital period

of a planet is directly proportional

to the cube of the semi-major

axis of its orbit. R 3 T 2 The radius of Moon around the Earth is 3.84x10^8m; the period is 8.61x10^4 s. A satellite is meant to go around the Earth everyday, which has a period 2.36x10^6. Therefore, scientists know the distance of the satellite to Earth's centre is 4.22x10^7m. When a planet is really far away from another planet, , distance becomes larger

so the force is approaching to zero. OGLE-2005-BLG-390Lb is known as the farthest planet, which is 21,500 light years from Earth, and it has almost zeron gravitational force between it and Earth. Fg from Earth Fg from Moon X R-X And object is floating between Earth and Moon in a weightless condition. The net force of two gravitational forces is zero. So Fg from Earth is equal to Fg from Moon. Fg moon=Fg earth

GMmoon(m)/(R-X)^2= GMmearth(m)/X^2

The mass of earth is 5.98x10^24kg

The mass of moon is 7.4x10^22kg

The distance between earth and moon is 3.8x10^8m from earth Example 1: g a *if a>g: occupants are floating towards the ceiling

*if a<g: occupants are pushed towards the floor

*if a=g: occupants are in apparent weightlessness Example 2: mg a in an elevator * if a>g: occupants are pushed towards the ceiling

*if a<g: occupants are pushed towards the floor

*if a=g: occupants are in apparent weightlessness Example 3: *if a>g: occupants are pushed towards the floor

*if a<g: occupants are pushed towards the ceiling

*if a=g: occupants are in apparent weightlessness a g

v= 2GM

d^1/2

v= (2x6.67x10^11x5.98x10^24

(6.38x10^8))^1/2

v=1.1x10^4m/s

This velocity applies to all bodies to escape from the earth because the formula doesn't relate to masses. if an object's velocity is greater or equal to escape velocity, the object moves constant or away from earth's orbit.

If the velocity is smaller than escape velocity, the object drops.

G, M, and R are constant, g.

Gravitational field strength equals the force on every Kg of mass. The mass of Mars is 6.4191 × 10^23kg, and the radius of Mars is 3.4x10^6m. So the gravity of Mars is 3.70 N/kg Explorer 1 The first satellite into the space from US. It was the third in World Record. The purpose of this satellite was to explore the completed Earth surface, but, actually, this satellite was one of products resulting from competition with Soviet Union. Sputnik 2 Was the second satellite from Soviet Union. It was the first spaceship which contained a living animal, Laika. Although Laika was burnt to death tragically in the cabin, she offered a great biological effort in astronomy which improved human's technology a lot. Laika was definitely a pivotal value to today's astronomy development. - Isaac Newton used 1. Kepler's laws, 2. centripetal force and 3. the Earth Moon system to summarize the formula of universal gravitation. Fc Fc between Earth and Moon:

=6.67x10^-11x5.98x10^24x7.4x10^22

(3.84x10^8)^2

=2.00x10^20 N F W= Fxr= Fg x r= x r

Work changes potential energy

W= Ep = Ef-Ei= 0- Ei= Ep is not zero at the surface of the earth, but at the infinite

Ep is the energy that the object is missing to escape from the attraction of the planet. This energy is also called Binding Energy. A 1200kg rocket needs 7.51x10^10 J

to escape from the attraction of the Earth. Ep = -6.67x10^-11 x5.98x10^24x120

6.37x10^6

= -7.51x10^10J In a circle:

The distance around a circle is 2πR, and the time once around the circle is a period, T. A satellite planet surrounds the Earth with a speed of 108m/s at the altitude of 2400km. By using two centripetal foce formulas, the period can be calculate without knowing

the mass of the satellite.

= 4π^2mR

T^2

T= 5.1x10^5 sec

Thus we can say that the satellite goes one lap around the earth for 6 days. Fp Ff A force exerts 10N on an object. There is a force of friction exerting 2N in the opposite direction. The net force on this object will be: Fnet =Fp-Ff= 10-2= 8N 1.2x10^5m/s 4.5x10^4m/s Potential Energy Conservation of Energy Orbital Period Orbit - is the curved path of an object around

a centre in space

- every planet has its orbit around.

- the seize of the orbit to attract other object depends on the mass of the planet - is the amount of time that a satellite needs to make a complete orbit around a centre The orbit around

the earth includes the

atmosphere above, thousands

of satellites, and the moon that

goes around the Earth everyday. For example, the orbital period of Moon is 27.3 days. And most of the communication satellites' periods are 24 hours. - is the energy of an object at a certain altitude(position) from the surface of the Earth.

- Ep= Fnet x d= m x a x d= m x g x h

- potential energy is a transferable energy A 2.5kg soccer ball is hanged on a 3m roof. The potential energy of this ball is Ep=mgh= 2.5x 9.81x 3= 74 J - it is a theory that states the transformation of energy. It says that, in a closed system, the amount of energy is constant no matter in mono or multiple states. No energy could be created or destroyed - Etotal= Ekinetic+Epotential+ heat by friction

Eki + Epi= Ekf+ Epf + heat = A ball slides down from

a certain height, 10m,

and when it is at 5.5m, it is 5m/s.

The total energy never change

as long as there's no heat produced.

Epi + Eki= Epf + Ekf + heat

10x9.81x m + mv^2

2

=m x9.81 x5.5+ mx5^2

2

the initial speed is 1.16m/s Kinetic Energy - a certain amount of energy that an object needs to accelerate from rest to a certain speed.

- E= F x d = m x a x d=

- Kinetic energy is a transferable energy A 1350kg truck accelerates to a certain speed, 55km/h. The kinetic energy in this truck is = 1350 x 15.3^2

2

= 1.58x10^5J Linear Acc. is the rate of the change in speed;

Centripetal Acc. is the rate of the change in direction Information and Image

Reference Google Image;

www.google.com

Wikipedia;

en.wikipedia.org The SI unit is m/s^2 After inserting Kepler's law: An object has to overcome the Earth's gravitational attraction, or the binging energy. So the object needs certain speed to escape in order not to fall down again.

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# Circular Motion and Gravitation

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