In the previous section, we saw the basics about SI system of units. In this section, we will see details about Length.
We will write it in steps:
1. We saw that in SI system, we measure length in metre.
(One metre is the distance traveled by light in vacuum during a time interval of 1/299,792,458 of a second).
Let us see some practical situations:
• Length of a study table is usually around 120 cm.
• The table that I am using right now, has a length of 122 cm.
• We can write it in three ways:
♦ Length of the table = 1.22 m.
♦ Length of the table = 122 cm.
♦ Length of the table = 1220 mm.
2. For objects like tables and chairs, the third method involving mm, is not generally used
• mm is used for smaller objects. Let us see an example:
• The length of the pencil that I am using right now, has a length of 14.5 cm.
• We can write it in 3 ways:
♦ Length of the pencil = 0.145 m.
♦ Length of the pencil = 14.5 cm.
♦ Length of the pencil = 145 mm.
• For objects like pens and pencils, the first method involving m, is not generally used.
• m is used for larger objects.
3. Now consider a small cylinder which has a length about the same as a pen.
• This cylinder is an important part of a precision instrument. It’s diameter is very important.
4. So a technician took an ordinary ruler and measured it’s diameter at one end.
• He got it as 2.7 cm. That is., 27 mm.
5. But this cylinder is to be used in a precision instrument. We want accurate diameter.
• Upon looking closely, the technician saw that the diameter is in between 27 mm and 28 mm.
6. But unfortunately, the smallest length which an ordinary ruler can measure is 1 mm.
• 1 mm = 0.001 m = 10-3 m.
• We say that the least count of that ruler is 10-3 m.
7. To measure lengths which are smaller than 1 mm, we need to divide each mm on that ruler, into 10 equal parts.
• That is not very convenient because, 1 mm is a small distance.
8. In such a situation, we use an instrument called the vernier calipers.
The smallest length which a vernier calipers can measure is 'one tenth of a mm'.
One tenth of a mm = $\mathbf\small{\frac{1}{10} \times 1 \,\text{mm}}$ = 0.1 mm.
• 0.1 mm = 0.0001 m = 10-4 m.
• So the least count of the vernier calipers is 10-4 m.
9. The technician used it on the cylinder and found that, the actual diameter is 27.3 mm.
• We know that: 27 < 27.3 < 28
• So greater precision is achieved.
10. In situations where still more precision is required, we use screw gauge or spherometer.
• They can measure lengths as small as 10-5 m.
11. Lengths smaller than 10-5 m are not uncommon in our day to day life.
• For example, municipality rules do not allow us to use plastic carry bags whose thickness is less than 50 microns.
(In some municipalities, all types of plastic carry bags are banned)
♦ Symbol for micron (micro meter) is 𝝻m.
♦ One 𝝻m = 10-6 m.
• In nano technology, scientists deal with objects which are as small as one nano meter.
♦ Symbol for nano meter is nm.
♦ 1 nm = 10-9 m.
12. To measure lengths smaller than 10-5 m, we use some special indirect methods. We will see them in later sections of this chapter.
Let us see some examples:
• Height of a two storeyed building used for residential purposes is around 6.4 m.
That is., 6 m plus 40 cm.
• A two storeyed building used for commercial or industrial purposes will have a greater height.
• Such heights can be measured using a measuring tape or simple survey instruments.
14. To measure the distance between two towns, we do not use m. Instead, we use km.
• Such distances can be measured using advanced survey techniques and satellite images.
15. To measure distances between planets and stars, we use astronomical unit.
♦ It’s symbol is au.
♦ 1 au is the mean distance between the center of earth and center of sun.
(Remember that, the path of earth is not circular. It is elliptical. So the distance from the sun is not a constant. That is why, we take mean distance).
♦ That distance is equal to 149597870700 m.
16. To measure distances between stars, we use light year.
♦ It’s symbol is ly.
♦ 1 ly is the distance traveled by light in 1 year.
(Remember that in 1 s, light travels a very large distance of 3 × 108 m).
♦ The distance traveled in one year is equal to 9.46 × 1012 km.
17. To measure even greater distances, we use parsec.
♦ It’s symbol is pc.
(We will see the definition of parsec in the next section).
♦ 1 pc is equal to 3.26 ly.
• Three examples of large distances are given below:
(i) The distance between earth and the moon.
(ii) The distance between earth and another planet.
(iii) The distance between earth and another star.
• In such cases, we cannot use a measuring tape. So we use indirect methods.
• Before we learn the details of those indirect methods, we must see some fundamental properties related to ‘large distances’. We will write them in steps:
1. In fig.2.3 below, a person PQ is moving in a car along a straight line.
• The top of a tree is marked as T. it is shown in green color.
• The tree is on the side of the road.
• When the man is at position A, his eye level is marked as PA
2. We know the science behind vision. Based on that, we can write:
The man sees the tree top T because the light rays reflected from T reaches his eyes.
• One such light ray is shown in magenta color. It is named as TPA
• Similarly, when the man is at position B, he sees T because of the ray TPB
• Note the angle between the two rays TPA and TPB. It is marked as θ1
3. Consider a hill on the side of the road. It’s top point is marked as H.
• The hill is behind the tree and is far away from the road.
• So H is far away from the man.
♦ When the man is at A, he sees H because of the ray HPA
♦ When the man is at B, he sees H because of the ray HPB
• The angle between those two rays is marked as θ2
4. We see that θ2 less than θ1
• This is because, H is further away from the man than T.
5. Consider a mountain on the side of the road. It’s 'snow covered top point' is marked as S.
• The mountain is behind the hill and is very far away.
• So S is very far away from the man.
♦ When the man is at A, he sees S because of the ray SPA
♦ When the man is at B, he sees S because of the ray SPB
• The angle between those two rays is marked as θ3
6. We see that θ3 is less than θ2
• This is because, S is further away from the man than H.
• The ray TPB is very different from TPA
• The man feels that, the tree moves past him in the opposite direction, at great speed.
8. The ray HPB is also very different from HPA
• But difference is not so high as in the case of the tree.
• The man feels that the hill moves past him in the opposite direction, but at a slow speed.
9. The ray SPB is also very different from SPA
• But difference is smaller than that in the case of the hill.
• The man feels that the mountain moves past him in the opposite direction, but at a very slow speed.
■ What happens if the distance is very large?
Ans: Obviously in that case, the movement will be so slow that, it will appear to be stationary. We experience such a situation in the case of the moon and the stars.
11. The height of Mount Everest is 8,848 m.
♦ That means, the tip of Mount Everest will be at a distance of 8,848 m from a person at ground level.
• The distance of the moon from the earth is 384,400,000 m.
♦ That means, the Moon will be at a distance of 384,400,400 m from a person at ground level.
12. Suppose that in the above fig.2.3, we mark the moon as M.
• The rays coming from M will be MPA and MPB
• M will be very high above S.
• So the angle θ between the two rays will be so small that, both rays will coincide.
• That means, the two rays are practically the same.
• So the moon will appear to be stationary.
13. But in our actual experience, we feel this:
■ The trees and hills are moving past us. But the moon is moving along with us. Why is that so?
Ans: At every point of our travel, practically the same ray from the moon reaches our eyes.
• So the moon is actually stationary with respect to us.
• But our minds know that we are moving.
• So we feel that, the moon is moving along with us.
(Suppose that we are moving in a very smooth train. Looking upwards through the sky roof, if we see only the moon and no trees,clouds, or any other objects, we will truly feel that the moon is stationary).
14. So we can conclude that, far away objects appear to be stationary when we are moving.
• We will be using this information while finding distances of far away objects.
We will write it in steps:
1. We saw that in SI system, we measure length in metre.
(One metre is the distance traveled by light in vacuum during a time interval of 1/299,792,458 of a second).
Let us see some practical situations:
• Length of a study table is usually around 120 cm.
• The table that I am using right now, has a length of 122 cm.
• We can write it in three ways:
♦ Length of the table = 1.22 m.
♦ Length of the table = 122 cm.
♦ Length of the table = 1220 mm.
2. For objects like tables and chairs, the third method involving mm, is not generally used
• mm is used for smaller objects. Let us see an example:
• The length of the pencil that I am using right now, has a length of 14.5 cm.
• We can write it in 3 ways:
♦ Length of the pencil = 0.145 m.
♦ Length of the pencil = 14.5 cm.
♦ Length of the pencil = 145 mm.
• For objects like pens and pencils, the first method involving m, is not generally used.
• m is used for larger objects.
3. Now consider a small cylinder which has a length about the same as a pen.
• This cylinder is an important part of a precision instrument. It’s diameter is very important.
4. So a technician took an ordinary ruler and measured it’s diameter at one end.
• He got it as 2.7 cm. That is., 27 mm.
5. But this cylinder is to be used in a precision instrument. We want accurate diameter.
• Upon looking closely, the technician saw that the diameter is in between 27 mm and 28 mm.
6. But unfortunately, the smallest length which an ordinary ruler can measure is 1 mm.
• 1 mm = 0.001 m = 10-3 m.
• We say that the least count of that ruler is 10-3 m.
7. To measure lengths which are smaller than 1 mm, we need to divide each mm on that ruler, into 10 equal parts.
• That is not very convenient because, 1 mm is a small distance.
8. In such a situation, we use an instrument called the vernier calipers.
The smallest length which a vernier calipers can measure is 'one tenth of a mm'.
One tenth of a mm = $\mathbf\small{\frac{1}{10} \times 1 \,\text{mm}}$ = 0.1 mm.
• 0.1 mm = 0.0001 m = 10-4 m.
• So the least count of the vernier calipers is 10-4 m.
9. The technician used it on the cylinder and found that, the actual diameter is 27.3 mm.
• We know that: 27 < 27.3 < 28
• So greater precision is achieved.
10. In situations where still more precision is required, we use screw gauge or spherometer.
• They can measure lengths as small as 10-5 m.
11. Lengths smaller than 10-5 m are not uncommon in our day to day life.
• For example, municipality rules do not allow us to use plastic carry bags whose thickness is less than 50 microns.
(In some municipalities, all types of plastic carry bags are banned)
♦ Symbol for micron (micro meter) is 𝝻m.
♦ One 𝝻m = 10-6 m.
• In nano technology, scientists deal with objects which are as small as one nano meter.
♦ Symbol for nano meter is nm.
♦ 1 nm = 10-9 m.
12. To measure lengths smaller than 10-5 m, we use some special indirect methods. We will see them in later sections of this chapter.
13. What we discussed above is related to small lengths. What about larger lengths ?
• Height of a two storeyed building used for residential purposes is around 6.4 m.
That is., 6 m plus 40 cm.
• A two storeyed building used for commercial or industrial purposes will have a greater height.
• Such heights can be measured using a measuring tape or simple survey instruments.
14. To measure the distance between two towns, we do not use m. Instead, we use km.
• Such distances can be measured using advanced survey techniques and satellite images.
15. To measure distances between planets and stars, we use astronomical unit.
♦ It’s symbol is au.
♦ 1 au is the mean distance between the center of earth and center of sun.
(Remember that, the path of earth is not circular. It is elliptical. So the distance from the sun is not a constant. That is why, we take mean distance).
♦ That distance is equal to 149597870700 m.
16. To measure distances between stars, we use light year.
♦ It’s symbol is ly.
♦ 1 ly is the distance traveled by light in 1 year.
(Remember that in 1 s, light travels a very large distance of 3 × 108 m).
♦ The distance traveled in one year is equal to 9.46 × 1012 km.
17. To measure even greater distances, we use parsec.
♦ It’s symbol is pc.
(We will see the definition of parsec in the next section).
♦ 1 pc is equal to 3.26 ly.
Measurement of large distances
(i) The distance between earth and the moon.
(ii) The distance between earth and another planet.
(iii) The distance between earth and another star.
• In such cases, we cannot use a measuring tape. So we use indirect methods.
• Before we learn the details of those indirect methods, we must see some fundamental properties related to ‘large distances’. We will write them in steps:
1. In fig.2.3 below, a person PQ is moving in a car along a straight line.
 |
| Fig.2.3 |
• The tree is on the side of the road.
• When the man is at position A, his eye level is marked as PA
2. We know the science behind vision. Based on that, we can write:
The man sees the tree top T because the light rays reflected from T reaches his eyes.
• One such light ray is shown in magenta color. It is named as TPA
• Similarly, when the man is at position B, he sees T because of the ray TPB
• Note the angle between the two rays TPA and TPB. It is marked as θ1
3. Consider a hill on the side of the road. It’s top point is marked as H.
• The hill is behind the tree and is far away from the road.
• So H is far away from the man.
♦ When the man is at A, he sees H because of the ray HPA
♦ When the man is at B, he sees H because of the ray HPB
• The angle between those two rays is marked as θ2
4. We see that θ2 less than θ1
• This is because, H is further away from the man than T.
5. Consider a mountain on the side of the road. It’s 'snow covered top point' is marked as S.
• The mountain is behind the hill and is very far away.
• So S is very far away from the man.
♦ When the man is at A, he sees S because of the ray SPA
♦ When the man is at B, he sees S because of the ray SPB
• The angle between those two rays is marked as θ3
6. We see that θ3 is less than θ2
• This is because, S is further away from the man than H.
7. So we find that, as the distance increases, the angle decreases.
• The man feels that, the tree moves past him in the opposite direction, at great speed.
8. The ray HPB is also very different from HPA
• But difference is not so high as in the case of the tree.
• The man feels that the hill moves past him in the opposite direction, but at a slow speed.
9. The ray SPB is also very different from SPA
• But difference is smaller than that in the case of the hill.
• The man feels that the mountain moves past him in the opposite direction, but at a very slow speed.
10. So we find that, when the distance increases, the 'movement of objects' in the opposite direction becomes slower and slower.
Ans: Obviously in that case, the movement will be so slow that, it will appear to be stationary. We experience such a situation in the case of the moon and the stars.
11. The height of Mount Everest is 8,848 m.
♦ That means, the tip of Mount Everest will be at a distance of 8,848 m from a person at ground level.
• The distance of the moon from the earth is 384,400,000 m.
♦ That means, the Moon will be at a distance of 384,400,400 m from a person at ground level.
12. Suppose that in the above fig.2.3, we mark the moon as M.
• The rays coming from M will be MPA and MPB
• M will be very high above S.
• So the angle θ between the two rays will be so small that, both rays will coincide.
• That means, the two rays are practically the same.
• So the moon will appear to be stationary.
13. But in our actual experience, we feel this:
■ The trees and hills are moving past us. But the moon is moving along with us. Why is that so?
Ans: At every point of our travel, practically the same ray from the moon reaches our eyes.
• So the moon is actually stationary with respect to us.
• But our minds know that we are moving.
• So we feel that, the moon is moving along with us.
(Suppose that we are moving in a very smooth train. Looking upwards through the sky roof, if we see only the moon and no trees,clouds, or any other objects, we will truly feel that the moon is stationary).
14. So we can conclude that, far away objects appear to be stationary when we are moving.
• We will be using this information while finding distances of far away objects.
In the next section, we will see Parallax method.
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