Force and Motion

Today we'll talk about

Force

Our first job is to define what we mean by force. Again, this is one of those words that has meaning in everyday language. Our meaning in physics does not have the same amount of ambiguity. A force is a push or a pull, generally by some observable contact between two objects. For example, we exert a force on a book when we push it with our hand. Whenever we see contact between two objects, we can assume there are forces between them. Unless we can see contact, there is no force except when we are dealing with the fundamental forces of nature, such as gravity, electric and magnetic forces. When those forces act, we can't see the contact between objects. In our modern perspective, even these forces result from "contact" between objects.

The SI unit of force is a Newton (abbreviated N), named for Sir Isaac Newton. Frequently, you'll encounter the English unit of force, called the pound (abbreviated lb). We'll talk more about that below.

Mass


What's the connection between force and motion? This question has been around for about as long as there have been people. It's only relatively recently that the correct answer was found; it wasn't until 1687, when Newton published the Principia, that the answer was given. Prior to that, nearly everyone else had it wrong! In fact, their wrong notions are quite similar to what you have "learned" in your direct experience, so it'll be hard for you to immediately drop your ~20 years of acquired experience and start using the proper relationship between force and motion.

There is an important preliminary to our discussion. That is the introduction of the physical quantity called mass. Mass measures the quantity of matter. The SI units of mass are kilograms (abbreviated kg). No, the English unit of mass is not the pound. That's a unit of force. In fact, the English unit of mass is the little used quantity known as the slug. It's so infrequently used, you need not worry about a conversion factor!

The amount of motion of an object is inversely proportional to its mass. What this means is the following. Suppose we subject different objects to an equal force, say 1 N. The amount of motion that those objects obtain from this force is smaller when the object is more massive. With an equal magnitude applied force, it's easier to move a ping-pong ball than it is to move a bowling ball!

Net Force


The wrong-headed notion that humanity had (and still has, unless they've had a good physics course!) is that the velocity of an object is proportional to the applied force. Why have so many people been wrong? The reason is that this conclusion is consistent with casual observations we make. When we push on an object it moves. If we push harder, it moves faster. If we stop pushing, the object ceases to move. Right?

The problem with such casual observations is that we are relating the force we apply to the object directly to its motion. We are ignoring other important forces on the object that affect it's motion. The most relevant, in this case, is friction.

To find the proper relationship between force and motion, we have to account for the total force on an object, a quantity in physics we call the net force. One aspect of force that we'll need to realize is that there are three things we needed to specify a force: first, a number (say 1); second, a unit (Newton); and third, a direction (north). Force, like velocity and acceleration, is an example of a vector quantity. Something which has a magnitude and a direction. Vector quantities are different from scaler quantities: things that have magnitudes only, such as mass and temperature. To find the net force, we have to add up quantities that have both magnitude and direction. For example, in a tug of war, if Joe's team pulls to the south with a force of 100 N and Sally's team pulls to the north with a force of 120 N, then the net force causing motion is directed to the north. Sally's team wins!

Newton's Laws


The importance that friction plays in making careful observations about the relationship between force and motion was first clarified by Galileo. He made observations about the amount of force necessary to keep an object moving with constant velocity (meaning, zero acceleration). He experimented with surfaces the object sat on, using successively smoother ones. He found that as the surface got smoother (friction was reduced) it took less and less force to keep an object moving with constant velocity. He then concluded that for an imaginary, perfectly smooth surface, an object would require no force at all to keep moving with constant velocity.

This is such an important fact, that Isaac Newton made it one of his three laws of motion, which I paraphrase below.

In the absence of a net force, an object moves with constant velocity.

It's not too hard to imagine now what a net force does to an object. When there is a net force, the object changes it's velocity: it accelerates. This is Newton's second law of motion, which we write in an equation as

F = ma

The F here refers to (net) force, m to the mass of the object and a to the acceleration of an object. The point is that an object's acceleration is proportional to the net force acting on it.

The most useful thing about Newton's second law is its predictive power. If we know two of the three quantities, F, m and a, then we can predict the third! Why this is so useful, is that if we know the net force on an object and its mass, we can find its acceleration. Knowing its velocity when the force starts being applied, we can find how its velocity changes while the force is acting. If we know its position when the force starts being applied, we can then find out where it is and how fast it is going at all other times. Pretty powerful stuff! This works so well, that NASA is able to predict exactly how to launch a satellite and land it on Mars. We'll see how all this works with some examples below.

With Newton's second law, we can find what the SI unit of force is in terms of fundamental quantities (being mass, length and time). Remember that the SI units of acceleration are m/s2 and that the units of mass are kg. That means that the SI units of force are kg.m/s2. So

1 N = 1 kg . m/s2

This is our first real example of the importance of expressing everything in SI units. We need to be sure forces are in Newtons, masses in kilograms and accelerations are in m/s2 before we apply Newton's second law to calculate one for the three quantities!

If this isn't confusing enough already, there is also Newton's third law that tells us where forces come from. The answer is that all forces come in action/reaction pairs. If my hand exerts a 5 N force on a book that points to the north (action), the book exerts a 5 N force on my hand that points to the south (reaction). If a horse exerts a 100 N force to the east on a cart (action), the cart exerts a 100 N force to the west on the horse (reaction).

Weight


We have already encountered the fact that (in the absence of air resistance), all objects fall to the earth with the same acceleration, g = 9.8 m/s2. Since an object must have mass, from Newton's second law, there must be a force on the object whose magnitude is mg. That force is called the weight of an object. What is this force? The answer is gravity. The earth exerts a gravitational force on all objects equal to their mass multiplied by the acceleration due to gravity (g=9.8 m/s2). Okay, so why aren't we always falling? Generally, when we are sitting, standing or even walking, there is a contact force (from a chair or the floor) that balances the force due to gravity. Hence, there is no net force on us in the vertical direction that would result in our accelerating in that direction.

It's important to not confuse the weight of an object (a force) with its mass (a measure of how much stuff there is in the object).

Examples


To illustrate the predictive power of Newton's second law, it is useful to do examples.

Example 1 An 50 kg object, initially at rest, is acted on by a 100 N net force pointing to the north. How fast is the object going after 5 seconds?

From Newton's second law, we predict that the object's constant acceleration is

a = F/m
a = 100 N / 50 kg
or, a = 2 m/s2

The direction of the acceleration is the same as the net force, to the north.

Armed with this knowledge, we can now use kinematics

delta_v = a x delta_t

The time interval (delta_t) here is 5 seconds, so

delta_v = (2 m/s2)(5 seconds) = 10 m/s

Since the initial velocity is zero, the object has a velocity of 10 m/s to the north after 5 seconds.

Example 2 A car slows down with constant acceleration when the brakes are applied. The net force that causes the car to stop is applied on it by the road. If that force has a magnitude of 2,000 N and it takes 5 seconds for the car to come to a complete stop when it is initially going 72 km/hr, what is the car's mass?
Our first job is to find the car's acceleration, but to use Newton's second law, we need to express it in SI units

delta_v = vf - vi = 0 km/hr - 72 km/hr = -72 km/hr
delta_v = (-72 km/hr)(1000 m/km)/(3600 s/hr) = -20 m/s

Now, since we have the velocity change in SI units, we can find the acceleration

a = delta_v/delta_t = -20 m/s / 5 s = -4 m/s2

OK so far? We know the net force on the car (by the road) and the car's acceleration. So its mass must be

m = F/a = 2,000 N / 4 m/s2 = 500 kg

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Last updated: 6 Sep 1999
Comments: bland@indiana.edu