9-20-99

Sections 4.1 - 4.5

We've introduced the concept of projectile motion, and talked about throwing a ball off a cliff, analyzing the motion as it traveled through the air. But, how did the ball get its initial velocity in the first place? When it hit the ground, what made it eventually come to a stop? To give the ball the initial velocity, we threw it, so we applied a force to the ball. When it hit the ground, more forces came into play to bring the ball to a stop.

A force is an interaction between objects that tends to produce acceleration of the objects. Acceleration occurs when there is a net force on an object; no acceleration occurs when the net force (the sum of all the forces) is zero. In other words, acceleration occurs when there is a net force, but no acceleration occurs when the forces are balanced. Remember that an acceleration produces a change in velocity (magnitude and/or direction), so an unbalanced force will change the velocity of an object.

Isaac Newton (1642-1727) studied forces and noticed three things in particular about them. These are important enough that we call them Newton's laws of motion. We'll look at the three laws one at a time.

The ancient Greeks, guided by Aristotle (384-322 BC) in particular, thought that the natural state of motion of an object is at rest, seeing as anything they set into motion eventually came to a stop. Galileo (1564-1642) had a better understanding of the situation, however, and realized that the Greeks weren't accounting for forces such as friction acting on the objects they observed. Newton summarized Galileo's thoughts about the state of motion of an object in a statement we call Newton's first law.

Newton's first law states that an object at rest tends to remain at rest, and an object in motion tends to remain in motion with a constant velocity (constant speed and direction of motion), unless it is acted on by a nonzero net force.

Note that the net force is the sum of all the forces acting on an object.

The tendency of an object to maintain its state of motion, to remain at rest or to keep moving at a constant velocity, is known as inertia. Mass is a good measure of inertia; light objects are easy to move, but heavy objects are much harder to move, and it is much harder to change their motion once they start moving.

A good question to ask is: do Newton's laws apply all the time? In most cases they do, but if we're trying to analyze motion in an accelerated reference frame (while we're spinning around would be a good example) then Newton's law are not valid. A reference frame in which Newton's laws are valid is known as an inertial reference frame. Any measurements we take while we're not moving (while we're in a stationary reference frame, in other words) or while we're moving at constant velocity (on a train traveling at constant velocity, for example) will be consistent with Newton's laws.

If there is a net force acting on an object, the object will have an acceleration and the object's velocity will change. How much acceleration will be produced by a given force? Newton's second law states that for a particular force, the acceleration of an object is proportional to the net force and inversely proportional to the mass of the object. This can be expressed in the form of an equation:

In the MKS system of units, the unit of force is the Newton (N). In terms of kilograms, meters, and seconds, 1 N = 1 kg m / s^{2}.

In applying Newton's second law to a problem, the net force, which is the sum of all the forces, often has to be determined so the acceleration can be found. A good way to work out the net force is to draw what's called a free-body diagram, in which all the forces acting on an object are shown. From this diagram, Newton's second law can be applied to arrive at an equation (or two, or three, depending on how many dimensions are involved) that will give the net force.

Let's take an example. This example gets us ahead of ourselves a little, by bringing in concepts we haven't talked about yet, but that's fine because (a) we'll be getting to them very shortly, and (b) there's a good chance you've seen them before anyway. Say you have a box, with a mass of 2.75 kg, sitting on a table. Neglect friction. There is a rope tied to the box and you pull on it, exerting a force of 20.0 N at an angle of 35.0¡ above the horizontal. A second rope is tied to the other side of the box, and your friend exerts a horizontal force of 12.0 N. What is the acceleration of the box?

The first step is to draw the free-body diagram, accounting for all the forces. The four forces we have to account for are the 20.0 N force you exert on it, the 12.0 N force your friend exerts, the force of gravity (the gravitational force exerted by the Earth on the box, in other words), and the support force provided by the table, which we'll call the normal force, because it is normal (perpendicular) to the surface the box sits on.

The free-body diagram looks like this:

We can apply Newton's second law twice, once for the horizontal direction, which we'll call the x-direction, and once for the vertical direction, which we'll call the y-direction. Let's take positive x to be right, and positive y to be up. The box accelerates across the table, so it has an acceleration in the x direction but not in the y direction (it doesn't accelerate vertically).

In the x direction, summing the forces gives:

The x-component of the force you exert is partly canceled by the force your friend exerts, but you win the tug-of-war and the box accelerates towards you. Solving for the horizontal acceleration gives:

a_{x} = 4.4 / 2.75 = 1.60 m/s^{2} to the right.

In the y direction, there is no acceleration, which means the forces have to balance. This allows us to solve for the normal force, because when we add up all the forces we get:

The gravitational force is often referred to as the weight. To remind you that this is actually a force, I'll generally refer to it as the force of gravity, or gravitational force, rather than the weight. The force of gravity is simply the mass times g, 2.75 x 9.8 = 26.95 N. Solving for the normal force gives:

F_{N} = 26.95 - 20.0 sin35 = 26.95 - 11.47 = 15.5 N. In many problems the normal force will turn out to have the same magnitude as the force of gravity, but that is not always true, and it is not true in this case.

A force is an interaction between objects, and forces exist in equal-and-opposite pairs. Newton's third law summarizes this as follows: when one object exerts a force on a second object, the second object exerts an equal-and-opposite force on the first object. Note that "equal-and-opposite" is the shortened form of "equal in magnitude but opposite in direction".

Consider the free-body diagram of the box in the example above. The box experiences 4 different forces, one from you, one from your friend, one from the Earth (the gravitational force) and one from the table. By Newton's law, the box also exerts 4 forces. If you exert a 20.0 N force on the box, the box exerts a 20.0 N force on you. Your friend exerts a 12.0 N force to the left, so the box exerts a 12.0 N force to the right on your friend. The table exerts an upward force on the box, the normal force, which is 15.5 N, so the box exerts a downward force of 15.5 N on the table. Finally, the Earth exerts a 26.95 N force down on the box, so the box exerts a 26.95 N force up on the Earth.

Although the forces between two objects are equal-and-opposite, the effect of the forces may or may not be similar on the two; it depends on their masses. Remember that the acceleration depends on both force and mass, and let's look at the force exerted by the Earth on a falling object. If we drop a 100 g (0.1 kg) ball, it experiences a downward acceleration of 9.8 m/s^{2}, and a force of about 1 N, because it is attracted towards the Earth. The ball exerts an equal-and-opposite force on the Earth, so why doesn't the Earth accelerate upwards towards the ball? The answer is that it does, but because the mass of the Earth is so large (6.0 x 10^{24} kg) the acceleration of the Earth is much too small (about 1.67 x 10^{-25} m/s^{2}) for us to notice.

In cases where objects of similar mass exert forces on each other, the fact that forces come in equal-and-opposite pairs is much easier to see.