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Introduction

The force of gravity is very familiar to us.  For example, it is the force that keeps us attached to Planet earth.  It is the reason why, if we throw a tennis ball up into the air, it comes back down to earth again.  It’s also responsible for the orbit of the moon around the earth, and the earth around the sun.  Providing an explanation of how the gravitational force arises is a complex and involved business.  Describing the effects of gravity and its impact on the various objects in the universe is not too difficult, but before we do so, it is necessary to introduce some preliminary ideas, to make the job easier. 

Newton’s Laws 

Newton’s laws provide the basis for evaluating the consequences of applied forces on objects, at least in the classical world.  In this context, the term “classical” can loosely be translated to mean not too fast, not too small or not too big.  In studying gravitational effects, Newton’s laws are valid, but general relativity provides a more exact description. 

Newton’s laws are: 

1.  A body continues in its state of uniform motion if the total force acting on the body is zero. 

2.  The rate of change of momentum of a body is equal to the total force acting on that body. 

3.  To every action there is an equal and opposite reaction. 

A fuller description of the laws can be found here. 

Centripetal Force 

In treating the effects of gravity, we are often interested in how the rotational motion of an object or objects is modified.  Consider a stone attached to a string, and suppose that we swing the stone in a horizontal circle (or nearly horizontal - the string requires some tension in the vertical direction to counteract the gravitational pull of the stone downwards).  Assume that the speed of rotation remains constant in time.  This situation is illustrated in the following figure.

We know from experience that swinging the stone in this manner sets up a tension force in the string, and that the faster we swing the stone, the greater the tension force.  It is this tension force that keeps the stone swinging in a circular motion.  If the string were to snap, then the tension force would become zero, and the stone would go flying off out of its circular orbit. 

Now, Newton’s second law tells us how to work out the tension force in the string.  To do so, we need to work out the rate of change of momentum of the stone in its circular orbit.  Since the mass of the stone remains constant in time, this problem reduces to working out the acceleration of the stone as it rotates on the string.

(It should be noted that even though the speed of the stone might not be changing, the stone is still subject to an acceleration because its direction of motion is changing.  Velocity, acceleration, force and momentum are what we call “vector” quantities, which means that they have direction as well as magnitude.  In the case of the stone on the string, the stone is accelerated because its velocity is changing direction, even though the magnitude of the velocity - the speed - remains constant.)

The magnitude of the acceleration can be worked out in a fairly simple way, by computing the rate of change of the velocity of the stone with time.  It can be shown that the magnitude of the acceleration of the stone is given by 

and the direction of the acceleration is along the string towards the centre of the circle of rotation. 

This acceleration is called the centripetal acceleration, and by using Newton’s second law, force = mass × acceleration, the centripetal force that is required to provide this acceleration is given by 

This is the force that is required to keep the stone rotating in its circular path on the end of the string.  Clearly, the centripetal force is supplied by the horizontal component T of the tension force in the string.  The following figure illustrates this.

An alternative means of expressing the centripetal force is obtained by noting that the speed v of the stone can be expressed in terms of the angular velocity ω through the relationship 

Substituting into the expression for centripetal force, we get 

We will be making use of this form of the centripetal force when we come to look at Kepler’s laws of planetary motion. 

Angular Momentum 

Angular momentum is an extremely important concept in many branches of physics, in particular quantum mechanics and atomic physics, where the angular momentum of the electrons in the atom determine many of the electrical and magnetic properties of the atom.  Angular momentum is also an important quantity in classical systems where rotational motion takes place. 

In the consideration of linear motion, momentum is defined as 

Where M is the mass of the object under consideration, and v is the speed of the object.  For rotational motion, the angular momentum is defined in an analogous fashion: 

Where I is termed the “moment of inertia” of the object under consideration and ω is the angular velocity.  Note that the moment of inertia of an object depends on the location of the axis about which the rotation is taking place, and is not an intrinsic property of the object itself. 

For the stone rotating on the end of a string (see the first figure of this section), the moment of inertia is given by

It can therefore be seen that the moment of inertia depends very strongly on how far away the mass lies from the axis of rotation.  The same remains true for solid objects that rotate about an axis that lies within the object mass. 

Angular momentum enters into the rotational equivalent of Newton’s second law.  The linear version states that 

Applied force = rate of change of (linear) momentum 

The rotational equivalent is 

Applied torque = rate of change of angular momentum 

Torque is equal to 

Where F is the force applied about an axis and r is the distance from the axis of rotation.  See here for a more detailed description of the concept of torque. 

The importance of angular momentum is that, like linear momentum, it is conserved if there are no external torques acting on the system.  Thus, the angular momentum of a dynamic system remains constant if there are no external torques acting on that system. 

Application:  Ice Skating

A nice example of angular momentum conservation is how ice skaters can increase their angular velocity of rotation by drawing in their arms.  We have often seen how a skater can rotate on the spot with arms outstretched, but when they draw their arms in, their rotational velocity increases.  This can be explained from conservation of angular momentum. 

We noted above that the moment of inertia of a rotating object (in this case the skater) depends on how far the mass of that object lies from the axis of rotation.  When the skater has his arms outstretched, his moment of inertia is high, because the mass of his arms lie some distance from the axis of rotation.  When he pulls his arms in, the moment of inertia decreases, because the mass of his arms is now much closer to the axis of rotation.  But the angular momentum J of the skater must remain constant, since there are no external torques acting on him.  The angular velocity ω must therefore increase, since the angular momentum is given by

and the moment of inertia I has decreased. 

We now have all the basic tools ready to discuss the effects of gravitation.  But before we can do this, we must discuss gravitation itself!

Next:  Gravity and Gravitation