Gravitation

Newton and Gravitation

SUMMARY

In 1687 Sir Isaac Newton first published his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) which was a radical treatment of mechanics, establishing the concepts which were to dominate physics for the next two hundred years. Among the book's most important new concepts was Newton's Universal Law of Gravitation. Newton managed to take Kepler's Laws governing the motion of the planets and Galileo's ideas about kinematics and projectile motion and synthesize them into a law which governed both motion on earth and motion in the heavens. This was an achievement of enormous importance for physics; Newton's discoveries meant that the universe was a rational place in which the same principles of nature applied to all objects. The Universal Law of Gravitation has several important features. First, it is an inverse square law, meaning that the strength of the force between two massive objects decreases in proportion to the square of the distance between them as they move farther apart. Second, the direction in which the force acts is always along the line (or vector) connecting the two gravitating objects. Moreover, because there is no "negative mass," gravity is always an attractive force. It is also noteworthy that gravity is a relatively weak force. Modern physicists consider there to be four fundamental forces in nature (the Strong and Weak Nuclear forces, the Electromagnetic force and gravity), of which gravity is the weakest. This means that gravity is only significant when very large masses are being considered.

Terms and Formulae

Terms

Universal Law of Gravitation - Newton's Universal Law of Gravitation states that

where m₁ and m₂ are the masses of any two objects under consideration and r₁ and r₂ are their respective position vectors.

Gravitational Constant - This the G that appears as a constant of proportionality in Newton's Universal Law of Gravitation. It has a value of 6.67×10^-11 Nm²/kg².

Formulae

Equation for the gravitational constant

g =

Newton's Law

Qualitatively Newton's Law of gravitation states that:

Every massive particle attracts every other massive particle with a force directly proportional to the product of their masses and inversely proportional to the square of distance between them

In vector notation, if is the position vector of mass m₁ and is the position vector of mass m₂, then the force on m₁ due to m₂ is given by:

= =

The difference of the two vectors in the numerator gives the direction of the force. The appearance of a cube, instead of a square, in the denominator is in order to cancel this direction-giving factor of | - | in the numerator.

Figure 1.1: Direction of force is the difference of the position vectors.

This force has some remarkable properties. First, we note that it acts at a distance , meaning that irrespective of any intervening matter, every particle in the universe exerts a gravitational force on every other particle. Furthermore, gravity obeys a principle of superposition. This means that to find the gravitational force on any particle it is necessary only to find the vector sum of all the forces from all the particles in the system. For example, the force of the earth on the moon is found by vector summing all the forces between all the particles in the moon and earth. This sounds like an immense task, but actually simplifies calculation.

Gravity as a central force

Newton's Universal Law of Gravitation produces a central force. The force is in the radial direction and depends only on the distance between objects. If one of the masses is at the origin, then () = F(r). That is, the force is a function of the distance between the particles and completely in the direction of . Obviously, the force is also dependent on G and the masses, but these are just constant--the only coordinate on which the force depends is the radial one.

It is easy to show that when a particle is in a central force, angular momentum is conserved, and motion takes place in a plane. First, let us consider the angular momentum:

= (×) = × + × = ×(m) + × = 0

The last equality follows because the cross product of with itself is zero, and since is entirely in the direction of , the cross product of these two vectors is zero also. Since angular momentum does not change over time it is conserved. This is essentially a more general expression of Kepler's Second Law, which we saw (here) also asserted the conservation of angular momentum.

At some time t₀, we have the position vector and velocity vector of the motion that define a plane P with a normal given by = ×. In the previous proof we showed that × does not change in time. This means that = × does not change in time either. Therefore, × = for all t. Since must be orthogonal to , it must always lie in the plane P.

Newton's Three Laws

Newton's Three Laws:

Newton's Three Laws, named after Sir Isaac Newton, who derived the laws, provide the basis for the study of Dynamics, and describe the fundamental laws of motion. These three laws will serve as a springboard for all other topics concerning dynamics.
Mathematically, the laws can be written as the following:

First Law: If F = 0 then a = 0 and v =constant
Second Law: F = ma
Third Law: F_AB = - F_BA

We will spend the nest three section of this SparkNote examining the ideas behind Newton's Three Laws, and explaining the derivation of their mathematical formulas...

The Concept of Force and Newton's First Law

Definition of a Force

Since force is the fundamental concept of Dynamics, we must give a clear definition of this concept before we proceed with Newton's Laws. A force is defined (very practically) as a push or a pull. Of course, we experience forces all the time in everyday lives. Whenever we lift something, push something or otherwise manipulate other objects, we are exerting a force. A force is a vector quantity, as it has both a magnitude and a direction. Let us show vector quality of a force practically: when exerting a force, for example pushing a crate, we can change the magnitude of our force by pushing harder or softer. We can also change the direction of our force, as we can push it one way or another. Since a force is a vector, all the rules of vector addition and subtraction, seen in Vectors apply. The vector quality of force allows us to manipulate forces in exactly the same way we manipulated velocity and acceleration in Kinematics.

With a formal definition of force, we can now examine its relation to motion through Newton's laws.

Newton's First Law

So how exactly does a force relate to motion? Intuitively, we can say that a force, at least in some way, causes motion. When I kick a ball, it moves. Newton makes this relation more precise in his first law:

An object moves with constant velocity unless acted upon by a net external force.

What does this mean? Let's start by looking at a special case where the constant velocity is zero, i.e. the object is simply at rest. Newton's First Law states that the object will stay at rest unless a force acts upon it. This makes sense: the soccer ball isn't going anywhere unless someone kicks it. This concept is true not only for v = 0, but for any constant velocity. Consider now a ball rolling with a constant velocity. Neglecting friction, the ball will continue to roll with the same velocity until it hits something, or someone kicks it. In physics terminology, it will keep the same velocity until acted upon by a net external force.

What does Newton mean by a net force? Consider a rope being used in a tug of war. There are definitely forces being applied to the rope but, if the two sides pull with the same force, the rope won't move. In this example, the two forces on the rope exactly cancel each other out, and there is no net force on the rope. It is thus possible for forces to act on an object, yet have the net force be zero. When evaluating the motion caused by forces acting on an object, remember to find the vector sum of those forces.

Also included in Newton's First Law, though not explicitly, is the concept of inertia. Inertia is defined as the tendency of an object to remain at a constant velocity. It is a fundamental property of all matter. In a sense, the idea of inertia is unnecessary; it just gives a name to the concept Newton describes in his First Law. However, you're bound to hear the word over and over in physics, so it is important to know to what it refers.

From our concept of inertia, we can develop the idea of an inertial reference frame, meaning a frame in which a body has no observed acceleration. This concept has limited application to classical mechanics, yet is essential for the study of Relativity. Consider a body with no net force acting upon it. For example, imagine yourself in an accelerating automobile. You look out the window, and the ground seems to be accelerating in a direction opposite the motion of the car. Clearly no net forces act upon the ground, yet from the frame of the car the ground is accelerating; in this case the car represents a non-inertial frame, and measurements of inertial fields from non-intertial fields do not conform to the rules of Newton's Laws. If, however, the car is traveling at a constant velocity, the ground will also appear to be moving back with a constant velocity. In this case, the frame of the car is inertial, as no net acceleration is observed. Any inertial reference frame is thus valid one in which to make calculations based on Newton's laws. Before we use these force laws, we must make sure we are making measurements from an inertial frame.

The concept of Mass and Newton's Second Law

Now we have both a definition of force, and a vague idea of how forces relate to motion. What we need is a precise way of relating the two. But even before we do this, we need to define another concept that plays a role in the relation between force and motion, that of mass.

Mass

Mass is defined as the amount of matter in a given body. This definition seems a little vague, and needs some explanation. Mass is a scalar quantity, meaning it has no direction, and is a property of the object itself, not its location. Mass is measured in kilograms (kg). Given a certain object, its mass will be the same on earth, on the moon, or in empty space. In contrast, the weight of the object in these different circumstances will change. We will explore further the relation between mass and weight when we have completed discussing Newton's laws. Yet even without a complete understanding of weight we can use weight to better understand the concept of mass. In our everyday experience, the heavier an object is (the more weight it has), the more mass it has. Thus our experience tells us that a baseball has more mass than a balloon, for example. As long as we do not think of them as the same concept, describing mass in terms of weight allows us to conceptualize mass in practical terms. From this concept of mass, we can more exactly relate force and motion.

Given a certain force, how does an object's motion correspond to its mass? Our intuition tells us that a more massive object moves slower if given the same force as a less massive object. We can throw a baseball with much greater speed than we can throw a massive ball of lead. Our intuition is correct, and is stated in Newton's Second Law.

Newton's Second Law

Newton's Second Law gives us a quantitative relation between force and motion:

secondlaw

F = ma

Stated verbally, Newton's Second Law says that the net force (F) acting upon an object causes acceleration (a), with the magnitude of the acceleration directly proportional to the net force and inversely proportional to the mass (m). Learn it and love it. Like it or not, this equation will be used at almost all times in virtually every physics course you take.

The Second Law relates two vector quantities, force and acceleration. Because both force and acceleration are vector quantities, it is important to understand that the acceleration of an object will always be in the same direction as the sum of forces applied to the object. The magnitude of acceleration depends on the mass of the object, but is always proportional to the force. Newton's Second Law gives an exact relation between the vectors force and motion. Thus we can use this law to predict the motion of an object given forces acting upon it, on a quantitative level.

Free Body Diagrams

The best method for calculating acceleration from force is through a free body diagram. This process, though fairly complicated, is extremely useful. We will go through it step by step:

• Step 1: Draw the physical situation in which an object exists. It may lie on an incline, be attached to a string, or simply be resting on the ground. Whatever the situation, draw it complete with any angles or distances that apply.
• Step 2: From the center of the body being examined, draw vectors representing each force acting upon the body, giving the magnitude of each one.
• Step 3: Sum all horizontal components of forces acting upon the object (this may require resolving a vector into its components (see Vectors).
• Step 4: Sum all vertical components of forces acting upon the object (using the same method as step 3).
• Step 5: Find the net force acting on the object, using the sum of the vectors found in steps 3 and 4.
• Step 6: Divide the net force by the object's mass to find the acceleration vector of the object.
• Step 7: From the acceleration vector, compute velocity, position, or any other necessary kinematic quantity.

There we have it! Finally, we can compute an exact relation between force and motion. With Newton's second law, we can take a given physical situation and find the acceleration, and thus the motion, of an object in the situation. In addition, using the method of free-body diagrams, we can evaluate any number of distinct forces. Such an ability is powerful, and will be used over and over in physics courses. We can now move on to Newton's Third Law, which further clarifies the nature of forces..

Newton's Third Law and Units of Force

Newton's Third Law

All forces result from the interaction of two bodies. One body exerts a force on another. Yet we haven't discussed what force, if any, is felt by the body giving the original force. Experience tells us that there is in fact a force. When we push a crate across the floor, our hands and arms certainly feel a force in the opposite direction. In fact, Newton's Third Law tells us that this force is exactly equal in magnitude and opposite in direction of the force we exert on the crate. If body A exerts a force on body B, let us denote this force by F_AB. Newton's Third Law, then, states that:

thirdlaw

FAB = - FBA

Stated in words, Newton's third law proclaims: to every action there is an equal and opposite reaction. This law is quite simple and generally more intuitive than the other two. It also gives us a reason for many observed physical facts. If I am in a sailboat, I cannot move the boat simply by pushing on the front. Though I do exert a force on the boat, I also feel a force in the opposite direction. Thus the net force on the system (me and the boat) is zero, and the boat doesn't move. We need some external force, like wind, to move the boat. Though this law seems obvious and unnecessary, we will see its importance when we apply Newton's laws.

Newton's third law also gives us a more complete definition of a force. Instead of merely a push or a pull, we can now understand a force as the mutual interaction between two bodies. Whenever two bodies interact in the physical world, a force results. Whether it be two balls bouncing off each other or the electrical attraction between a proton and an electron, the interaction of two bodies results in two equal and opposite forces, one acting on each body involved in the interaction.

Amazingly enough, Newton's Three Laws provide all the necessary information to describe the motion involved in any given situation. We will soon study the applications of Newton's Laws, but we first need to take care of the units of force.

Units of Force

The unit of force is defined, quite appropriately, as a Newton. What is a Newton in terms of fundamental units? Given that acceleration (a) = m/s² and mass = 1 kg, we can find out from Newton's Second Law: F = ma implies that a Newton, N = kg (m/s²) = (kg ƒ m)/s². Therefore, one Newton causes a one kilogram body to accelerate at a rate of one meter per second per second. Our definition of units becomes important when we get into practical applications of Newton's Laws.

Summary of Newton's Laws

We can now give an equation summary of Newton's Three Laws: First Law: If F = 0 then a = 0 and v =constant
Second Law: F = ma
Third Law: F_AB = - F_BA

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