Mechanics Gravity. Lana Sheridan. Nov 29, De Anza College

Transcription

1 Mechanics Gravity Lana Sheridan De Anza College Nov 29, 2018

2 Last time angular momentum of rigid objects conservation of angular momentum examples

3 Overview Newton s law of gravitation gravitational field gravitational potential energy

4 Motion of the stars, planets, and falling apples Part of Newton s genius was to realize that the same force that dictates the motion of the stars and planets is what holds us on the Earth. This realization is called the Newtonian synthesis. The planets are also falling, but they are constantly falling around the Sun.

5 Gravitation Newton s Universal Law of Gravitation states that any two massive object in the universe interact with each other according to the same rule. Newton s Law of Universal Gravitation F G = Gm 1m 2 r 2 for two objects, masses m 1 and m 2 with a distance r between their centers. G = Nm 2 kg 2.

6 r to generate a value for G., each of mass m, fixed to the er Gravitation or thin metal wire as illusf mass M, are placed near the larger spheres causes the rod brium orientation. The angle eam reflected from a mirror.1 is often referred to as an e varies as the inverse square examples of this type of force in vector form by defining a rected from particle 1 toward is (13.3) 5 k/x, where k is a constant. A direct pro- Figure 13.1 Cavendish apparatus for measuring gravitational forces. S F 21 m 1 Consistent with Newton s S S third law, F 21 F 12. rˆ12 r S F 12 m 2 Figure 13.2 The gravitational force between two particles is attractive. The unit vector r^12 is directed from particle 1 toward particle 2. r 2 ˆr 1 2 F G,1 2 = Gm 1m 2 for two objects, masses m 1 and m 2 at a distance r. G = Nm 2 kg 2.

7 The Universal Gravitational Constant, G F = G m 1m 2 r 2 G sets the scale of the force due to gravity (and makes the units come out correctly). G = N m 2 /kg 2

8 The Universal Gravitational Constant, G F = G m 1m 2 r 2 G sets the scale of the force due to gravity (and makes the units come out correctly). This could also be written: so it is quite a small number. G = N m 2 /kg 2 G = N m 2 /kg 2

9 The Universal Gravitational Constant, G The fact that G is so small indicates that gravity is a weak force. Force Rel. strength Range (m) Attract/Repel Gravitational attractive Electromagnetic 10 2 attr. & rep. Weak Nuclear < attr. & rep. Strong Nuclear 1 < attr. & rep.

10 The Universal Gravitational Constant, G The fact that G is so small indicates that gravity is a weak force. Force Rel. strength Range (m) Attract/Repel Gravitational attractive Electromagnetic 10 2 attr. & rep. Weak Nuclear < attr. & rep. Strong Nuclear 1 < attr. & rep. Despite the fact that gravity is a weak force, it is the only one that (typically) matters on large scales.

11 Acceleration due to Gravity This force in that it gives objects weight, F g. For an object of mass m near the surface of the Earth: and where F g = mg g = GM E R 2 E M E = kg is the mass of the Earth and R E = m is the radius of the Earth. The force F g acts downwards towards the center of the Earth.

12 Acceleration due to Gravity The acceleration due to gravity, g, can vary with height! F G = GM ( ) Em GME r 2 = m r 2 = mg Depends on r the distance from the center of the Earth. Suppose an object is at height h above the surface of the Earth, then: g decreases as h increases. g = GM E (R E + h) 2

13 Acceleration due to Gravity The acceleration due to gravity, g, can vary with height! F G = GM ( ) Em GME r 2 = m r 2 = mg Depends on r the distance from the center of the Earth. Suppose an object is at height h above the surface of the Earth, then: g decreases as h increases. g = GM E (R E + h) 2 g is the not just the acceleration due to gravity, but also the magnitude of the gravitational field.

14 Fields field A field is any kind of physical quantity that has values specified at every point in space and time.

15 Fields field A field is any kind of physical quantity that has values specified at every point in space and time. Fields were first introduced as a calculation tool. A force-field can be used to identify the force a particular particle will feel at a certain point in space and time based on the other objects in its environment that it will interact with. We do not need a description of the sources of the field to describe what their effect is on our particle. Gravitational force: Electrostatic force: F G = m( GMˆr r 2 ) = mg F E = qe

16 Fields Gravitational force: Electrostatic force: F G = m( GMˆr r 2 ) = mg F E = qe Gravitational field: g = F G m Electric field: E = F E q The field tells us what force a test particle of mass m (in the gravitational case) or charge q (in the electrostatic case) would feel at that point in space and time.

17 e perpendicular to the electric field lines passing through them. t the end of Section 25.2, the equipotential surfaces associated ectric Examples field consist of of Fields a family of planes perpendicular to the 25.11a shows some representative equipotential surfaces for this ld produced f charge Fields are drawn with lines showing the direction of force that a test particle will feel at that point. The density of the lines at that point in the diagram indicates the approximate magnitude of the A spherically symmetric electric field produced by a point charge force at that point. An electric field produced by an electric dipole tential surfaces (the dashed blue lines are intersections of these surfaces with the page) and elecq S E b c

18 Examples of Fields The gravitational field caused by the Sun-Earth system looks something like: 1 Figure from

19 nce of the field and measure oting Gravitational the force exerted Field of on the it. Earth ject (in this case, the Earth) rce that would be present if a a b Near the surface of the Earth: le in a field analysis model. in an area of space in which d a property of the particle, l version of the particle in a ational, and the property of mass m. The mathematical icle in a field model is Equa- (5.5) e particle in a field model. In lts in a force is electric charge: ure 13.4 (a) The gravitational were placed in the field. The magnitude of the field vector at any location is the magnitude of the free-fall acceleration at that location. Farther out from the Earth: a

20 Gravitational Field of the Earth a Uniform g: e particle establishes a gravitad by measuring the force on a e a particle of mass m is placed at it experiences a gravitational b (5.5) gure 13.4 (a) The gravitational s ld theory vectors of gravitation in the in vicinity of a S g m S S F g mg continued A test mass m experiences a force F g = mg, where g is the field vector.

21 Gravitational Potential Energy The gravitational potential energy stored by two masses at a distance r from each other is given by: Choosing U(r) = 0 at r =. U(r) = Gm 1m 2 r This will always be a negative number.

22 Gravitational Potential Energy (13.14) center of the Earth particles inside the U is always negative th system, a similar is, the gravitational es m 1 and m 2 sepa- (13.15) for any pair of par- 1/r 2. Furthermore, and we have chosen finite. Because the do positive work to y the external agent Gravitational potential energy of the Earth particle U(r) = Gm system 1m 2 r O GM E m R E U M E Earth The potential energy goes to zero as r approaches infinity. R E Figure Graph of the grav- r

23 which is the energy equivalent of 89 gal of gasoline. NASA engi craft as it ejects burned fuel, something we have not done her effect of this changing mass to yield a greater or a lesser amoun Example using Grav. PE: Escape Speed How fast does an object need to be projected with to escape Earth s gravity? Escape Speed R E S v f S v i m 0 h r max Suppose an object of mass m i with an initial speed v i as illust to find the value of the initial tance away from the center of system for any configuration. the Earth, v 5 v i and r 5 r i 5 R v f 5 0 and r 5 r f 5 r max. Beca these values into the isolated- Solving for v i 2 gives 1 2 M E Figure An object of mass m projected upward from For a given maximum altitud required initial speed. We are now in a position t speed the object must have a

24 Escape Speed The object begins with kinetic energy K = 1 2 mv i 2 energy U = GM E m R E. and potential To escape Earth s gravity well the object needs to reach U = 0. Trade kinetic for potential energy. Find an expression for the speed the object must have at the surface of the Earth to escape.

25 Escape Speed The object begins with kinetic energy K = 1 2 mv i 2 energy U = GM E m R E. and potential To escape Earth s gravity well the object needs to reach U = 0. Trade kinetic for potential energy. Find an expression for the speed the object must have at the surface of the Earth to escape. system: object + Earth, isolated K + U = 0

26 Escape Speed The object begins with kinetic energy K = 1 2 mv i 2 energy U = GM E m R E. and potential To escape Earth s gravity well the object needs to reach U = 0. Trade kinetic for potential energy. Find an expression for the speed the object must have at the surface of the Earth to escape. K + U = 0 (0 1 ( ( 2 mv i 2 ) + 0 GM )) E m = 0 R E 1 2 mv i 2 = GM E m R E 2GM E v i = R E

27 Summary Newton s law of gravitation gravitational field gravitational potential energy Homework Ch 13 Ques: 1; Probs: 1, 5, 17, 19, 29, 33