PY1008 / PY1009 Physics Gravity I

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1 PY1008 / PY1009 Physics Gravity I M.P. Vaughan

2 Learning Objectives The concept of the centre of mass Fundamental forces Newton s Law of Gravitation Coulomb s Law (electrostatic force) Examples of Newton s Law of Gravitation

3 Centres of mass The centre of mass of an object is an average of its position, weighted by its density distribution. For example, for a sphere of constant density, the centre of mass is the geometric centre.

4 Newton s Law of Gravitation Consider two bodies with a distance r between their centres of mass* r body body 1 *The centre of mass of an object is an average of its position, weighted by its density distribution. For a sphere of constant density, it is the geometric centre.

5 Newton s Law of Gravitation The gravitational force the bodies exert on one another is given by F Gm1m r, where G is the universal gravitational constant and m 1 and m are the gravitational masses of each object. G m kg -1 s -.

6 An inverse squared law Note that the gravitational force between two bodies is inversely proportional to the square of the distance between them. F Gm1m r, F 1 r. This is known as an inverse square law.

7 An inverse squared law This means that, for instance, if the distance r is doubled, the force is reduced by a factor of 4. Starting with F 0 1 r 0, we put r 1 r 0. So F F0 r r

8 Gravitational v inertial mass The gravitational masses determine the strength of the gravitational force. This is a different concept to inertial mass, which is a measure of a body s reluctance to being accelerated. Hence the use of the Old English font to discriminate the two.

9 The Equivalence Principle According to the Equivalence Principle, due in its original form to Galileo, gravitational and inertial mass are equivalent. That is m m. The equivalence principle was taken by Einstein to be a tenet of general relativity. Galileo Galilei

10 The Equivalence Principle Assuming that the Equivalence Principle holds, Newton s Law of Gravitation may then be written as F Gm1m r. This has the same logical form as before but is now given in terms of the inertial masses rather than the gravitational masses.

11 The strength of the gravitational force The gravitational force acts between all objects that have mass. r However, it is only when an object has a very large mass (such as a planet) that the gravitational force becomes significant.

12 The strength of the gravitational force Consider the gravitational attraction between two people As a rough estimate, take r m 1 m and r 0.5 m m 1 m 70 kg.

13 The strength of the gravitational force Inserting these values into F Gm1m r we have, to 3 significant figures, F N m kg s 0.5 m -6 kg m s kg70 kg,,

14 The strength of the gravitational force Compare this to the force (i.e. weight) due to the Earth felt by either person. m R E M E

15 The strength of the gravitational force The gravitational force between the man and the Earth is therefore m F GM R E E m, R E M E where m is the mass of the man, M E is the mass of the Earth and R E is the radius of the Earth.

16 The strength of the gravitational force The mass and radius of the Earth are M E kg and R E m. so F E N, N 687 N.

17 The strength of the gravitational force The weight of the man due to the Earth s gravitational force is therefore about times greater than the force he feels towards his spouse. In words, his weight is five hundred million times the gravitational force between him and his wife.

18 Gravitational force near a planet s surface We have just seen that the gravitational force acting on a mass m on the Earth s surface is given by F GM R What is the force acting on a mass m a height h above the surface? Putting E E R E h, m. we have F R E GM m E 1 /. h R E

19 Gravitational force near a planet s surface Now if h is very much smaller than R E, i.e. h / R E 1, then this term is negligible and may be ignored. This means that F GM R E E m.

20 Gravitational force near a planet s surface It will turn out to be very useful to put GM E g, R where g (or little g ) is (roughly) constant (RE will vary a little at different locations because the Earth is not a perfect sphere). The gravitational force can then be written E F mg.

21 Gravitational force near a planet s surface Comparing F mg to Newton s Second Law F ma, we see that g must have the dimensions of acceleration. In fact, g is known as the acceleration due to gravity.

22 Gravitational force near a planet s surface Comparing F mg to Newton s Second Law F ma, we see that g must have the dimensions of acceleration. In fact, g is known as the acceleration due to gravity.

23 The value of g We have defined g to be g GM R E E. Using the values of M E and R E given earlier, g ms, 9.80 ms.

24 Variations in g In fact, the gravitational force at the surface of the Earth varies due to various reasons. Firstly, the Earth is not a perfect sphere, so R E is not a constant R E (Greatly exaggerated for clarity)

25 Variations in g Secondly, there may be large variations in the local density Region of high/low density

26 Variations in g Thirdly, the rotation of the Earth can change the apparent value of g due to centrifugal effects* *To be discussed in a later lecture

27 The Equivalence Principle again Consider two objects with different masses m 1 and m dropped from the same height h. The forces on each one are given by F1 m1a 1 m1 g F ma mg Dividing the first equation by m 1 and the second by m, we see that a and 1 a g In other words, both objects will have the same acceleration...

28 The Equivalence Principle again Since both objects have the same acceleration, (in the absence of any other force) they will fall at the same rate. This is a particular case of an alternative expression (due to Galileo) of the equivalence principle In the absence of any other forces, all objects subject to the gravitational force will fall at the same rate.

29 Using the kinematic equations with g Note that, since g is constant, we may use it in place of a in the kinematic equations derived earlier. That is, we can put v u gt, x 1, gt ut x 0 v u gx x. 0 Note that since g acts downwards, we have written it here as being negative.

30 Freefall time of fall Let us consider an object of mass m in free-fall. Suppose the object starts from a height h from rest y h a = -g If the only force acting on the object is gravity, we can use the kinematic equations to obtain y 1 h gt. 0

31 Freefall time of fall The object will reach ground level at y = 0 when So, rearranging 1 gt h. t h g. Remember, that in the absence of any other force, this result is the same for an object of any mass.

32 Freefall height of flight Consider an object fired vertically with an initial velocity u. What will be its maximum height? y We use v u g x x 0 to obtain u v u gy. 0

33 Freefall height of flight When the object reaches its maximum height, we will have v = 0, so u gy 0. So the maximum height must be y u g.

34 Air resistance So why, in practice, do objects fall at different rates? This is due to air resistance. Let us suppose that the force is proportional to the velocity squared F R v.

35 Air resistance We can write this as F R a R v. where a R is the constant of proportionality. The forces acting on the object may be represented by a force diagram

36 Force diagrams In the present case, we have The gravitational force acts downwards F R mg Air resistance acts upwards in opposition to the velocity

37 Terminal velocity Applying Newton s Second Law, we have F ma arv mg. As the object gets faster, the air resistance increases until the net force is zero and there is no further acceleration. At this point a R v mg 0, so v This is known as the terminal velocity. mg a R.

38 Terminal velocity Inspecting the expression for the terminal velocity, we see that now it does depend on the mass. v mg a R If we had two objects with the same coefficient a R but of different mass m 1 and m, we would find. v v 1 m m 1. Hence the object with the greater mass would reach a greater terminal velocity.

equivalence principle gravitational mass is equivalent to inertia

39 Summary Newton s Law of Gravitation This is an inverse squared law, relating the force between gravitational masses According to the equivalence principle gravitational mass is equivalent to inertia Another statement of the equivalence principle is that all bodies fall at the same rate

40 Summary Newton s Law of Gravitation Close to a planet s surface, the gravitational force may be taken to be constant In the absence of other forces, objects in freefall close to a planet s surface may be modelled using the kinematic equations In practice, air resistance cannot be neglected

CHAPTER 10 GRAVITATION

CHAPTER 10 GRAVITATION Earth attracts everything towards it by an unseen force of attraction. This force of attraction is known as gravitation or gravitation pull. Universal Law of Gravitation:- Every