Before you read this, I suggest you read post 16.13.
In the previous post (post 16.49), we saw the difference between forces that stretch things and forces that act in the opposite direction to compress things. In this post we will think about forces that act in any direction.
Let’s suppose we want to use a rope to pull a cart along a straight railway track. We know, intuitively, that to get the maximum effect, we need to pull in the direction of the track – in the picture the angle θ (the Greek letter “theta”) needs to be 0o. If we pull at right-angles to the track (θ= 90o), our pulling has no effect. As the angle increases from 0o to 90o, the effect of pulling becomes less.
If F represents the applied force, the force that is effective in moving the cart is given by F multiplied by the cosine of θ; we write this effective force as F cos θ and call it the component of the force in the direction of the track.
The table shows the value of the cosine of some angles. If you want to find the cosine of a different angle, you can use a scientific calculator, use the cosine calculator at http://www.rapidtables.com/calc/math/Cos_Calculator.htm or use a spreadsheet like Excel. (If you use the COS function in Excel, you will need to divide the angle, measured in degrees, by 180 and multiply by 3.14159; the reason for this is that Excel doesn’t measure angles in degrees but uses an alternative unit called the radian.) To find out more about cosines, read “Some geometry” at the end of this post.
The angle at which we pull the cart can be greater than 90o. The picture above shows the angles that the coloured lines make with the black line and the graph below shows the values of cosines for angles in the range 0o to 360o. Notice that for angles that are greater than 90o but less than 270o, the cosine is negative because the effect of the force is to pull the cart in the opposite direction.
So the effect of a force depends on its direction, as well as on its numerical value, measured in newtons (see post 16.13 for information on the units, newtons, we use to measure force). We call something which is specified by direction as well as by a number a vector; the number is called the magnitude of the vector. So we can say that a vector has both magnitude and direction. An ordinary number that has magnitude only is called a scalar. The component of a vector (see above) is its effective value in a defined direction – so it can be represented solely by a number and is a scalar. If we want to make it clear that a force, F, is a vector, we write it in bold as F; in handwriting we usually underline it.
Force isn’t the only concept that’s a vector – we’ll meet more of them in a later post.
17.3 Three-dimensional vectors
17.4 Displacement, velocity and acceleration
17.13 Centripetal force – throwing the hammer
17.36 More about work – line integrals
17.37 More about work – cross-products of vectors
20.1 Mathematical proof and science (for appendix 1)
Appendices: Some geometry
1. Proof of Pythagoras’ theorem
We have seen that a scientific experiment can prove that an idea is false but not that it is true (see post 16.3).
Mathematical proofs are completely different. They develop ideas that must follow logically if we start by making certain assumptions – called axioms. The ideas that we develop in this way are called theorems.
Pythagoras’ theorem concerns right-angled triangles. If we agree on what we mean by a right-angle (a quarter of a complete revolution which means 90o) and what we mean by a triangle (a figure drawn on a plane that has three sides), we can show that Pythagoras’ theorem must follow logically.
The picture shows three right-angled triangles that all have sides of the same length: a (the shortest), c (the longest) and b (length less than c). They are drawn in different colours so that we can see what happens when we rearrange them.
On the left-hand side, they have been fitted into a square of side (a + b). The way this has been done leaves a white square of side c. The area of this square is c × c which we write as c2.
On the right-hand side, the four triangles have been fitted into a square of side (a + b) in a different way. This way leaves a small white square (top left-hand side of the square) of side a; the area of this square is a2. It also leaves a larger white square (bottom left-hand side) of side b; the area of this square is b2. The sum of the areas of these two white squares is a2 + b2.
The white areas in both boxes must be equal – because they’re what’s left when the four triangles have been fitted into the square of side (a + b). So
a2 + b2 = c2.
We’ve proved Pythagoras’ theorem!
There are many ways of proving Pythagoras’ theorem; I think this is the simplest.
2. Cartesian coordinate systems
The picture above shows a horizontal line, that starts at a point O, and a vertical line, that starts at the same point. They are like the axes of a graph. We can specify the position of a point, like P, by the distance, x, we need to travel along the horizontal line and the distance, y, that we need to travel along the vertical line in order to reach it from O. Then we call O the origin, the horizontal line the X-axis and the vertical line the Y-axis of a Cartesian coordinate system. We call it a “Cartesian” coordinate system because this way of doing geometry was developed by the French philosopher and mathematician René Descartes. Because the X and Y-axes are perpendicular, changing the distance along the X-axis does not change the distance along the Y-axis. Similarly, changing the distance along the Y-axis does not change the distance along the X-axis. So we call this system for specifying the position of P an orthogonal Cartesian coordinate system. (Not all Cartesian coordinate systems need be orthogonal – we often use non-orthogonal coordinates to specify positions of atoms in crystals.)
3. Definition of cosine and sine
We will represent the length of the line OP by L and the angle it makes with the X-axis by θ (the Greek letter “theta”). Then cos θ (the cosine of θ) is defined by
x= L cos θ.
Here L cos θ means L multiplied by cos θ. Also sin θ (the sine of θ) is defined by
y= L sin θ.
We call x the projection of L on the X-axis and y the projection of L on the Y-axis. So the cosine and sine of an angle are a means for calculating projections.
You may have learnt rules for defining cosine and sine at school that involved the sides of right-angled triangles. You may have been even more unlucky and been taught the non-existent English word SOHCAHTOA to help you remember them. My advice is to forget what you were taught and remember what I’ve written here; it makes it much easier to use cosines and sines in practice. The stuff about triangles is a throwback to the ancient Greek way of doing geometry. Fortunately, Descartes made our lives easier!
4. Relationship between cosine and sine
We can think of the picture as showing a right-angled triangle with sides whose lengths are x, y and L. Then according to Pythagoras’ theorem (see above)
x2 + y2 = L2.
In mathematics, x2 simply means x multiplied by itself. The definitions of cosine and sine give us ways of calculating x and y from L and θ. We can use them to write Pythagoras’ theorem in the form
(L cos θ)2 + (L sin θ)2 = L2.
The brackets show that we multiply everything inside the brackets by itself. We can write this equation in the form
L2(cos θ)2 + L2(sin θ)2 = L2.
(If you’re not sure about this step, note that (5 ×7)2 = 52 ×72; 5 × 7 =35 and 35 × 35 = 1 225; 52 =25 and 72 = 49, and 52 ×72 = 25 ×49 = 1225.) We can divide both sides of this equation by L2 to get
(cos θ)2 + (sin θ)2 = 1.
(If you’re not sure about this step, suppose the cost of 5 apples and 5 bananas is 100 cents. If we represent the cost of an apple by A and the cost of a banana by B, we can express this statement as the equation 5A + 5B = 100 cents. We can get the cost of one apple and one orange by dividing both sides of the equation by 5 so that A + B = 20 cents.)
We usually write (cos θ)2 as cos2θ and (sin θ)2 as sin2θ, so that the equation above is usually written as
cos2θ + sin2θ = 1.
The cosine of 60o is exactly 0.5, so sin2(60o) = 1 – (0.5 × 0.5) = 1 – 0.25 =0.75. Then sin(60o) = 0.866 because 0.866 × 0.866 = 0.75.
5. Definition of tangent
We define tan θ (the tangent of θ) by tan θ= sin θ/cos θ (its sine divided by its cosine).
In the previous section we saw that cos(60o) = 0.5 (exactly) and that sin(60o) = 0.866, so tan(60o) = 0.866/0.5 = 1.73.
6. Cosine and sine waves
A picture in the main post (above) shows coloured lines that make different angles with a black line. We can think of the positions of these coloured lines as being generated by a line that has its origin where the coloured lines intersect. If we rotate this line anti-clockwise, it generates increasingly bigger angles. If it starts at the position of the black line, its starting angle is 0o. As it rotates, it goes through the positions of the 45o line, the 90o line, the 135o line and so on, until it reaches its original position – it has then rotated 360o (it’s gone round a whole circle). So the position of the line is the same when it has rotated 360o + θ as when it had rotated by θ. So cos(360o + θ) is the same as cos θ and sin(360o + θ) is the same as sin θ. The position of the rotating line continues to repeat itself every 360o, as do its cosine and sine.
So graphs of cos θ and sin θ, plotted against θ, look like repeating waves (cosine and sine waves) as shown in the graph below. Notice that the two ways can be generated from each other by shifting one of them by an angle of 90o.