Measuring Large Distances

In this chapter, We will study about various methods used to measure large distances such as Echo method, triangulation method, and parallax method.

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Summary

1. To measure large distances between objects methods such as echo method, the laser method, sonar method, radar method, triangulation method and Parallax method are used.
2. Measuring long distance by echo method involves sending sound wave and receiving the echo back. The time taken for the sound to reach the top of the hill and to come back as echo is used in the measurement of distance. Distance of the object, s = v x t /2 where v is the velocity of sound wave and t is the time taken.
3. Parallax is the displacement of an object because of a change in the observer’s point of view. This effect is used to measure the distance of nearby stars. [latex]D=\frac b\theta[/latex] where b is arc length, θ is the angle between two observing points, D is the distance between objects.
4. Long distance can also be measured by Kepler’s third law. If T1 and T2 and the periods of revolution of two planets 1 and 2 and a1 is the known radius of the orbit of planet1, then a2, the unknown radius of the orbit of planet2 is given by a2 = a1 x (T2 / T1) ^ (2/3)
5. Measuring small distance such as radius of an atom can be done using Avogadro’s hypothesis. The radius of the atom is given by r = (M / 2πNρ) ^ (1/3) where N is Avogadro’s number, M is the molecular weight and ρ is the density of the object.

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1. Measuring Large and Small Distance
2. Parallax Method
3. Example of Parallax Method
4. Distance of Planet from Earth
5. External References

How do we normally measure length? Normally the range of the length which we see around us falls under this range that is 10 to the power minus 3 to 10 to the power 2 meters.  Any length that falls within this range is generally measured using a metre scale which we normally see in our labs.

Measuring Large Length and Smaller Distances

For length which is outside this range segments for lengths greater than 10 to the power 2 meter, we use a method called parallax method.  Apart from parallax method, there are many other methods such as echo method, a laser method, sonar method, radar method and triangulation method. We will discuss parallax method in detail below. Similarly, or extremely small lengths that are below the range of 10 to the power minus 3 we use methods like electron microscope or tunneling microscopy.

An example of very small lengths is the size of a molecule. The size of a molecule is so small that you cannot even see it with your naked eye. To measure the size of a molecule you cannot use a meter scale. Moreover, the normal microscopes which we use don’t have that much of resolution to measure for such small lengths. We generally use special microscopes that are the electron microscopes or sometimes the tunneling microscopes.

Parallax Method

We will discuss the parallax method in detail. What is parallax method?. Before we discuss what is this method all about, we should first know what is the meaning of the word parallax. Parallax is the effect whereby position or direction of an object appears to differ when viewed from different positions. When I say parallax it means that we are viewing the same object but from two different positions.

Parallax Method Example

The best example to understand this is by close your left eye and right eye to observe things. Let us suppose you have some object say you have a candle which is quite a far from you. You first close your right eye and view the candle with your left eye.  Then you open your right eye and view the candle by closing your left eye and try to view the same candle which is located at the same position.

You will find that there is a change in position.  Just try this as I keep saying, close your left right I try to view an object which is at some distance from where you are.  Then try to view the same object closing your left eye and you would see that there is a slight change in the position of the object. Why did this happen?. This is because you observed the same object from two different positions. So, the distance between these two different observation points is known as the basis and this phenomenon is known as parallax.

Distance between Ronnie and Tree

In our example, the distance between the left eye and the right eye is the basis. The distance between your left eye and right eye is the basis. The phenomenon that when you close your left eye you can see the object position slightly changed as when you close your right eye this entire phenomenon is referred to as parallax.

Let us suppose Ronnie is standing at some place and wants to estimate the distance of the tree which is very far away from him. It is not possible to measure this distance using a meter scale because this distance is greater than 100 meters. We will employ parallax method here and we will do the same experiment as I told you before. He would observe it closing his left eye then he would observe it closing his right eye.  Let us suppose R is the position of his right eye and L is the position of his left eye.

Mathematical Modelling

The distance between the two eye is your basis. We denote this basis as b. Now, You want to measure the distance of the tree from Ronnie. The angle which the two points of observation subtend at the tree is theta. Let us say the distance between the tree and the left eye of Ronnie is D.

We know that this D is very much greater than this small b.  The distance between the two eye is very small when compared to the distance of the tree from Ronny. It means that b is very much less than D. We can say that B by D is much less than one. The angle, theta, which is subtended here is an extremely small angle. If we assume that O is the center of a circle, then D becomes the radius of the circle and this L and R can be considered as an arc of the circle. With these assumptions, we can write that theta is equal to L R by D that is equal to B by D.

Formula to Find Distance

[latex] \theta=\frac{LR}D\;=\;\frac bD[/latex]

[latex]D=\frac b\theta[/latex]

From this we can write, the capital D is equal to b by Theta now using this relation we can determine the value of capital D right.

In this experiment, We observed that a tree was located far from this guy Ronnie.  First, he closed his left eye and observed the tree using his right eye. Then he closed his right eye and observed the tree with his left eye. Now, you would ask me practically how will you determine the angle theta. How is it possible to practically determine the angle theta while you are closing your left eye and observing the tree by your right eye?.  You need to measure this angle using a protractor. If you know the basis d and the angle, theta you can very easily calculate the distance of the tree from Ronny. This is how we apply parallax method for calculations of distant objects.

Distance of Planet From Earth

We will see another example here now let us suppose we have to find in the distance of a distant planet from Earth. We observe the planet from Earth from two different observation points. Let us say we take by an observation point as A (taken during June), the other observation point as B(taken during December). We observe the planet from these two points.

Since the object is very far off so this object can be treated as a small point and the distance between the points A and B are the basis. The basis, in this case, is nothing but the diameter of the earth. If we know the angle, that is the angle subtended at this planet from both the observation points then we can find out the value of D. The value of D would be equal to b by Theta so this relation will be helpful in finding the distances of distant objects.

External References

1. Reflection and Triangulation Method For Measuring Large Distances
2. Measuring Distance using Parallax Method by NASA

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