Basic Physics Of Quantum Theory, The. Basil S Davis
of the planet from the sun. So Newton was able to conclude that the force between two objects of masses m1 and m2 at a distance of r from each other is proportional to the product m1m2 and inversely proportional to r2. So this proportionality can be written as
A proportionality relation can be written as an equation by introducing a constant factor. So Newton’s Law of Gravitation is fully written as
where G is called the Universal Gravitational Constant. Scientists use the Syst`eme Internationale or SI system for defining the units of measurement. This system is also called the MKS system because its units are meter, kilogram and second.3 G = 6.67 × 10−11 in SI units.
What is the gravitational force acting on an object of mass m near the surface of the earth? Let us call the mass of the earth M. Let the radius of the earth be R. Consider an object a short distance above the ground, where by short distance we mean not more than about 10 kilometers. This is small compared to the radius of the earth which is about 6370 km. The force exerted by the earth on this object of mass m is therefore
where the distance between the mass and the earth is taken to be the distance between the mass and the center of the earth, which is R.
If this object were dropped from a small height above the ground, it would fall with an acceleration which we shall call g that is given by Newton’s Second Law as F = mg. Therefore
Cancelling m from both sides, we obtain
The mass of the earth M = 5.97 × 1024 kg, the radius of the earth R = 6.37 × 106 m and G = 6.67 × 10−11 in SI units. Putting these numbers into the equation, we obtain the acceleration g of a falling object to be 9.8 m/s2. So the force of gravity acting on an object of mass m is given by mg. We call this force the weight of the object in a technical sense. So a mass of 10 kg would have a weight of 98 newtons (written 98 N) where the newton is the SI unit of force.
Exercise 2.3.
(a) Earlier you found the acceleration of the earth as it orbits the sun. Knowing the distance of the earth to the sun and the value of G find the mass of the sun. (Hint: See Eq. (2.2). Here g is the acceleration of an object in the gravitational field of the earth. Replace this by the acceleration of the earth in the gravitational field of the sun.)
(b) The mass of the moon is 7.35×1022 kg and the radius of the moon is 1726 km. Find the acceleration with which an object would fall due to the force of gravity close to the surface of the moon.
(c) If an object is fired horizontally close to the ground it could go into orbit round the earth if the speed of the object is sufficiently high. Using the fact that the centripetal acceleration of an object moving along a circle is v2/r find the minimum horizontal velocity v with which a rocket must be fired for it to go into stable orbit round the earth. For simplicity assume the earth is a smooth sphere with no hills, trees, buildings or birds, and ignore air resistance.
2.6Work and energy
When a force acts on an object and displaces the object through a distance, we say that the force performs work on the object. If the displacement is exactly along the direction of the force, the work done is simply the product of the force and the displacement. If the force is at an angle to the displacement, the work done is less than the product of force and displacement. If the force is at right angles to the displacement, the work done is zero. Mathematically, we say work W = FS cos θ where θ is the angle between the force of magnitude F and the displacement of magnitude S.4
When an apple falls from a tree, work is done by the force of gravity on the apple. Because the force of gravity is in the same direction as the displacement the work done is positive. If the apple drops through a vertical height h, the work done by gravity on the apple is mgh.
What is the effect of this work done by the force of gravity? The immediate effect is that the apple speeds up. A moving object is said to have kinetic energy by virtue of its motion. If the mass is m and the speed is v the kinetic energy of the object is given by
Prior to the apple’s fall it was at some height above the ground. If this height is h meters, then we say that it had a potential energy of mgh. As it falls, its potential energy changes into kinetic energy and work is being done on the apple by the force of gravity.
The Work-Kinetic Energy Theorem states that the work done by a force on a body is equal to the change of kinetic energy of the body.
Potential and kinetic energy are forms of mechanical energy. We see that potential energy can be converted into kinetic energy. The reverse happens when we throw an object upwards. In the next chapter we shall see that there is another kind of energy called heat energy, and that it is possible to convert heat into mechanical energy and vice versa. So there is a universal law — called the Law of Conservation of Energy — which states that energy cannot be created or destroyed, but may be converted from one form to another. In SI units energy is measured in joules (J).
Exercise 2.4.
(a) What is the kinetic energy of the earth as it orbits the sun, ignoring the kinetic energy due to the rotation of the earth about its axis?
(b) 50,000 kg of water drop every second down a waterfall through a height of 30 m. If 80 per cent of this gravitational energy could be converted to electricity, how much electrical energy can be produced at the bottom of the fall per second? Energy produced per second is called power and expressed in units called watts (W).
2.7Determinism
A major consequence of Newtonian physics was that it created a sense of absolute universalism in the minds of many people. According to the laws of Newton every object follows a deterministic behavior. For example, consider two spheres moving along a straight line towards each other and moving away from each other along the same straight line after collision. If we know the masses and velocites of the spheres before they collide, we can predict the velocities of the spheres after they collide, assuming the total kinetic energy to be unchanged in the collision. In the following chapter we shall see how we came to know that all matter is made of atoms and molecules. If it were possible to measure the masses, positions and velocities of every single atom and molecule in the universe at a given moment of time, then it is theoretically possible to predict the exact configuration of all the atoms and molecules an hour later, a day later, a year later, a billion years later. This means that the exact state of the universe a billion years from now has already been determined. Nothing can change the flow of the history of the universe. Whether you will remain alive and if so what color clothing you will be wearing on a day exactly 10 years from today has already been determined. There is no such thing as free will because the circuits of the brain follow the laws of physics and whatever they do is dictated by necessity and they cannot do anything other than what is determined by the laws. So free will is an illusion. When we think we are choosing something over another we are simply moving in the direction determined by the laws of physics. Ironically, this sort of scientific determinism leads to conclusions very similar to nonscientific beliefs