Universal gravitation

Let’s look at this pretty boy and this pretty girl …

Let’s look at this pretty boy and this pretty girl … Do they exert a force on one another?

Do they exert a force on one another? They’re not in contact… But they ARE exerting a force on one another… What is that force?

Newton’s Law of Gravity There is a force of attraction (GRAVITATIONAL FORCE) between each pair of objects in the universe that is proportional to the masses of the objects inversely proportional to the distance between them

Newton’s Law of Gravity For two masses m1 and m2: Fon 2, due to 1 Fon 1, due to2 m1 m2 SAME MAGNITUDE, OPPOSITE DIRECTIONS!

Newton’s Law of Gravity The magnitude of the gravitational force exerted by m2 on m1 is: The direction of the force is always towards the other particle – the force is ATTRACTIVE!

Let’s go back to the pretty boy and the pretty girl r=2m mB=72kg mA=54kg

What is the gravitational force exerted by: The pretty boy on the pretty girl? The pretty girl on the pretty boy?

We know that (by Newton’s 3rd Law), the MAGNITUDE of these forces are the SAME, but OPPOSITE in DIRECTION. FA on B FB on A

The magnitude is given by EXTREMELY SMALL!

Gravitational Force So we see that for small masses, gravitational attraction is quite negligible (but it’s there). However, if at least one of the masses is large (like a planet, for example) gravitational attraction is LARGE!

Gravitational Force For large masses, r is taken to be the center-to-center distance. That is, we treat them like PARTICLES

The Earth and the Moon What is the magnitude of the gravitational force exerted by the Earth on the Moon? (mmoon=7.36 x 1022kg, mEarth=5.98 x 1024kg, center-to-center distance from Earth to moon = 3.84 x 108m)

The Earth and the Moon LARGE!

Mass and weight Mass and weight are two intrinsically different notions. The mass is a measurement of how much matter is in an object, while weight is a force, and measures how hard gravity is pulling on that object. A better scientific definition of mass describes it as having inertia, which is the resistance of an object to being accelerated when acted on by an external force. Your mass is the same wherever you are because the amount of stuff you're made of doesn't change, but your weight depends on how much gravity is acting on you at the moment;

Mass and weight The mass is measured in kilos or (pounds) and the force is measured in Newton. A Newton is the force that would give a mass of one kilogram an acceleration of one meter per second per second.

Mass and weight A bathroom scale measures … The weight or the mass?

Mass and weight THE WEIGHT! A bathroom scale measures … The weight or the mass? THE WEIGHT! Not the mass, because it measures the effect of the gravity over your mass.

The bathroom scale should give us a reading in Newton, rather than in kilos, but it is convenient to identify weight and mass in most applications.

Weight on Earth Your weight is just the force exerted by the earth on you. What is the weight of an object with mass m on the surface of the Earth?

Weight on Earth But we know that your weight is just equal to so

Weight a distance d from the surface of the Earth What if mass m is a distance d from the surface of the Earth? (common scenario: airplanes, mountain climbing, etc.) Then the distance between the center of the Earth and the mass would change. Gravitational force on mass m, and hence, its weight will change!

Weight a distance d from the surface of the Earth A mass m is a distance d away from the surface of the earth. So the weight of mass m (=gravitational force exerted by earth on mass m) is

Weight a distance d from the surface of the Earth So the weight of mass m (=gravitational force exerted by earth on mass m) is SMALLER THAN THE WEIGHT ON THE SURFACE OF THE EARTH!

Weight a distance d from the surface of the Earth Weight varies inversely with the square of the distance from the earth’s center m W(N) m W= mgo r= RE 700 600 500 400 r > RE 300 200 100 r (x 106 m)

Weight a distance d from the surface of the Earth Problem: At what distance above the surface of the earth is the acceleration due to gravity 0.980 m/s2 if the acceleration due to gravity at the surface has a magnitude of 9.80 m/s2?

Weight a distance d from the surface of the Earth From Newton’s Law of Gravity: You can plug in the values of G, mEarth, RE, and solve for r’, but you also know that, at the surface of the Earth (1) (2)

Weight a distance d from the surface of the Earth Divide (2) by (1): (3) (2)

Weight a distance d from the surface of the Earth Solving for r’: But r’ is the distance from the CENTER (NOT the SURFACE) of the Earth to mass m: (3) (2)

Weight Elsewhere The mass of Venus is 4.87 x 10ˆ24 kg, and its radius is 6,051,000 m. (a) What is the acceleration due to gravity on the surface of Venus? (b) What is the weight of a 5.00 kg rock on the surface of Venus?

Weight Elsewhere Plug in the data … and get (a) What is the acceleration due to gravity on the surface of Venus? Plug in the data … and get

Weight Elsewhere (b) What is the weight of a 5.00 kg rock on the surface of Venus?

Mass and weight So, to summarize: Your mass is the same wherever you are, but your weight depends on how much gravity is acting on you at the moment.

Multiple Objects Three particles are arranged as shown. What is the net gravitational force that acts on particle A due to the other particles? mB 2.0cm Particle Mass, kg mA 6.0 mB 4.0 mC FBA 4.0cm FCA mC mA

So what do we do with the forces???? We must add them as vectors!!! Multiple Objects So what do we do with the forces???? We must add them as vectors!!! Particle Mass, kg mA 6.0 mB 4.0 mC mA mB mC 4.0cm 2.0cm FBA FCA

Multiple Objects mA mB mC 4.0cm 2.0cm FBA FCA

Multiple Objects Plugging in the numerical values, we get:

Multiple Objects mB FBA 2.0cm 76 104 4.0cm FCA mC mA

Summary To apply Newton’s Law of Gravity: EVERYTHING’S HERE! What are the masses involved? What is the distance between them? (remember, we need the CENTER-TO-CENTER distance) EVERYTHING’S HERE!

Summary The acceleration due to gravity near the surface of the Earth

How was the constant G measured? Click here to see how http://www.youtube.com/watch?v=4JGgYjJhGEE&feature=related

Henry Cavendish was born on 10 October 1731 in Nice, France, where his family was living at the time. The Cavendish experiment, performed in 1797–98 was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant.[

The apparatus constructed by Cavendish was a torsion balance made of a six-foot wooden rod suspended from a wire, with a 2-inch diameter 1.61-pound (0.73 kg) lead sphere attached to each end. Two 12-inch 348-pound lead balls were located near the smaller balls, about 9 inches away, and held in place with a separate suspension system. The experiment measured the faint gravitational attraction between the small balls and the larger ones.

Planet Mass (kg) Radius (m) Mercury 3.30 x 10ˆ23 2,440,000 Venus 4.87 x 10ˆ24 6,051,000 Earth 5.97 x 10ˆ24 6,378,000 Moon 7.35 x 10ˆ22 1,738,000 Mars 6.42 x 10ˆ23 3,397,000 Jupiter 1.90 x 10ˆ27 71,492,000 Saturn 5.69 x 10ˆ26 60,268,000 Uranus 8.66 x 10ˆ25 25,559,000 Neptune 1.03 x 10ˆ26 24,764,000 Pluto 1.31 x 10ˆ22 1,160,000