Chapter 3 Class Notes
This week, we introduce the concept of Force. Forces cause objects to accelerate and decelerate. Another way of thinking of it is that any object that is accelerating or decelerating must be subject to a force, or combination of forces.
For example, when a car slows down, the force causing the car to decelerate is the frictional force between the wheels and the brakes, and between the tires and the road. If the car is driving on ice, then there is no friction, and the car continues forward even when you step on the brakes. This is what Sir Isaac Newton had in mind when he composed his second law of motion;
“Objects continue in uniform motion, in a straight line, unless acted upon by an external force”
The fact of the matter is that here on earth, there are always external forces present, usually in the form of friction, that always causes objects to deviate from uniform motion, in a straight line, and so the full import of Newton’s second law is hard for us to appreciate, at first.
The one place where objects do continue in uniform motion, in a straight line, is up in space where there is no friction, because spacecraft are moving around in a vacuum, plus, the effects of the gravitational force are diminished, for reasons that we will discuss in the next chapter.
The units of Force
Physicists determined, by experiment, that the acceleration, a, imparted to an object is proportional to the applied force, F,
written mathematically as
F a a where a is the “proportional to” sign
What this means is that if you apply a bigger force to the object, the object acquires a proportionally bigger acceleration. Furthermore, if you plot a graph of applied force, F, and the resulting acceleration, a, you would get a straight line.
If you then measure the slope of the straight line, you get a number that is the constant of proportionality between the force and the acceleration, and we call that number the mass of the object, m, sometimes called the inertial mass. The inertial mass is a measure of an objects resistance to move.
The complete equation, then, that relates force, F, and mass, m, and acceleration, a, is quite simple, it goes like this
F = ma
which is just the equation of a straight line graph with, F, plotted on the y axis and, a, plotted on the x-axis yielding a straight line, with a slope, m, passing through the origin.
Now, if the force is equal to the product of the mass and acceleration, then the units of the force must be equal to the units of mass times acceleration, which is kg ms-2. This unit is also known as the Newton (N) to commemorate Sir Isaac Newton.
Force as a Vector
We use vectors to describe the direction and the size of the force, and the resulting acceleration.
In this chapter, all the problems are one dimensional, which means that the forces will always be parallel to each other, but not necessarily in the same direction, for example,
Here, we have two forces, F1 and F2 pointing in opposite directions. In this case, I have indicated that F1 is larger than F2 and so the object accelerates from left to right. The way to write this result mathematically, however, is to choose an x-y coordinate system that is aligned with the direction along which you wish to find the acceleration, for example,
Now, you simply find the resultant force vector in the direction of + x, which means that you also need a sign convention so that you know which way + x is,
Now we have everything we need to solve this problem.
By summing the forces we obtain,
+F1 + - F2 = m a
So, you see now that the difference of two forces can also give rise to an acceleration. If we invent some numbers, let F1 = 10 N and F2 = 5 N, and a 2 kg mass, taking the difference, we get
+10 - 5 = 2 a
Thus, we quite easily find that
a = + 2.5 m/s2
where the plus sign means the object is accelerating in the +x direction.
Forces at an Angle
There are a set of interesting problems where the force is acting at some angle, like a block on an inclined plane, or a jet in a steep climb;
To solve this type of problem, you just rotate the x - y coordinate system so that one of the axes is again aligned with the direction of the acceleration vector, like this
and then solve for the acceleration in the +x direction like before. All we have done is assigned the +x direction to be “up the inclined plane” or “in the direction of the jet thrust vector”.
One more refinement. If you do not like drawing blocks or jets, you can use the “Center of Mass” symbol to represent an object, like this,
This type of drawing is called a “free body diagram”.
The Difference Between Weight and Mass
An objects weight, W, is a consequence of Earth’s gravitational force acting on the objects mass, m, and is calculated using
W = mg
where, g = 9.8 m/s2 , is the acceleration imparted to all objects free-falling under the influence of Earth’s gravity.
If you don’t want an object to fall, you have to provide a supporting surface, like a table for example, which provides an equal and opposite force, called the normal force, N, that holds the object up.
N = W exactly. N is an example of a reaction force, the table “reacts” to produce exactly the right amount of force needed to balance the weight of the object. It is the electrostatic forces between the atoms and molecules of the table material that produces the reaction force. When the table is unable to generate sufficient reaction force, as when a very large weight is placed on the table, the table breaks and the object accelerates, at g, to the next supporting surface, usually the floor!
When the table does support the weight, the object is not accelerating since N = W. If you resolve the forces, N - W = 0, which means that the product, ma = 0, but since the mass, m, is not zero, that means the acceleration, a, must be zero, and so the object is just sitting there quietly, not accelerating.
Gravitational and Inertial Mass
Before we move on, I want to enlighten you as to one of the many bizarre coincidences in nature, and that is the equivalence of inertial and gravitational mass. As already described, the inertial mass is determined by experiment, by measuring the slope of the F vs. a graph.
The forces, F, are contact forces, that is, someone or something is pushing on the block to make it accelerate, but gravity is not like that. Gravity is a force that acts at a distance.
The Earth revolves around the Sun, but there is no rope connecting the two. Somehow, that gravitational force is communicated across the vacuum of space.
Now, given the completely different nature of contact forces and forces acting at a distance, there is absolutely no reason to expect that they should both yield the same result for the mass, m, but they do. Thus, the gravitational mass equals the inertial mass, which is very convenient since we can quickly determine the masses of objects simply by weighing them.
Think of the alternative if this coincidence turned out not to be true. There would presumably be “Mass shops” which is where you would go if you wanted to know the mass of something. You would walk into the shop, hand the shop-keeper the object, he or she would then apply a known force to it, measure the resulting acceleration and calculate the mass for you. It would all be very tiresome.
Elevators
Getting back to forces, you now know that not only does a force give rise to an acceleration, but also the difference of two or more forces can give rise to an acceleration, which is why you get that funny sinking feeling when you go down in an elevator, or the sensation of feeling heavier when you go up.
Elevators are boxes pulled by cables. You stand in the box, the cable pulls on the box, the floor of the box pushes you up with a force N, and the difference between that force and your weight force, W, is equal to your acceleration, a.
Before we can solve this problem, we need a coordinate system and a sign convention. I always choose the + direction to be the same as the direction of the acceleration vector, then there are no minus signs to worry about. Plus, I’ve only chosen a one-dimensional coordinate, y, since were only interested in accelerations up and down.
Next, resolve, or equivalently, add, the force vectors in the direction of the acceleration vector, so,
N - W = ma
Now N is the force provided by the supporting surface, remember? Well, that would also be the force registered on a bathroom weigh scale if the person were standing on one, because it would be the bathroom scale that is supporting the person at that point. So, if we wish to find the apparent weight of the person as they are accelerating up in the elevator, what we really want to solve for is N, so, rearranging, we get
N = ma + W
Now you see that the person apparently weighs more than their gravitational weight, W, by an amount equal to ma, when the elevator accelerates up.
We could redraw the whole thing for the elevator accelerating down, but all that would change is the direction of the acceleration vector. So, now resolving the forces again in the +y direction, we get
N - W = -ma
which solving for the apparent weight yields
N = -ma + W
so, you see that the person apparently weighs less than their gravitational weight, W, by an amount equal to ma. See?
So, when you go up you feel a bit heavier, and when you go down you feel a bit lighter, but the sensation lasts only while the elevator is accelerating. Usually, after a couple of seconds, the elevator stops accelerating since it’s reached it’s maximum operating velocity.
Weightlessness
There something else you should know about elevators. If the cable snaps (God forbid!) then you accelerate downwards at a = g right? So, re-writing the previous equation we get
N = -mg + W
but mg is W, so actually what you have is
N = -W + W
which of course means N = 0. So the bathroom scale reads zero which means you are weightless; a consequence of free-falling in Earths gravity! What’s happening is the bathroom scale is constantly accelerating just out of reach of your tootsies, even though it may be located just under your feet!
Blocks pulled by Blocks
An interesting set of problems involves blocks connected by strings pulling each other along rather like a train.
There are 3 blocks of different mass connected by two strings. The combined system is pulled by a third string with a force P. Now, a tension force, T, exists within each of the two strings connecting the blocks. For example, consider block 1, it is being pulled by the tension T1. That same tension is pulling back on block 2, because the tension is the same at all points along the string. If the tension were not the same, different parts of the string would be accelerating at different rates and would be jiggling around, which is not what’s going on. The whole system is, in fact, accelerating steadily with a uniform, or constant acceleration, a, like this
We are trying to find the acceleration, a, of the system.
The next step is to consider the forces acting on each block individually, but before we begin, we need a sign convention and a coordinate system.
Lets consider Block 3 first, resolving forces we find
Block 3
P - T2 = m3 a
Next Block 2,
Block 2
T2 - T1 = m2 a
And lastly, block 1
Block 1
T1 - 0 = m1 a
If the masses, m1 , m2 , and m3 and the force P are known, then what we have above are three equations and three unknowns; the unknowns being the two tensions and the acceleration a. These three equations can be solved simultaneously, and the easiest way to do that is to add them up. The merit of doing this is that the unknown tensions all go away,
P - T2 = m3 a
+ T2 - T1 = m2 a
+ T1 - 0 = m1 a
______
P = ( m1 + m2 + m3 ) a
______
So, the result,
P = ( m1 + m2 + m3 ) a
yields the acceleration if the force P and the three masses are known. Lets make up some numbers. Let m1 = 1 kg, m2 = 2 kg and m3 = 3 kg, and let the pulling force P be 10 N. Then, rearranging the equation for a, we have
P
a = ______= 10 = 10 = 5 ms-2
1 + 2 + 3 6 3
( m1 + m2 + m3 )
13