Scalars and Vectors
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If someone ask you a Questions, ‘How far is Mumbai?’ to locate Mumbai on a map. What answer will satisfy him? You will need to know the distance in Kilometers and the direction.
But if I ask what is the temperature of the object? Or how many people live there. you need only one number to get the full information. The distance to Bombay is not complete information without giving the direction of that measurement. But the temperature, say 35° C is complete and unambiguous information. Temperature is called Scalar (independent of direction). The quantities like distance needs the direction to become complete are called Vectors(dependent of direction).
The 3-D Space All the objects in the universe have 3 dimensions; they have length, breadth and height. A line has only one dimension-length. An area has two dimensions i.e. breadth and length. A solid object has 3 dimensions. (ref. fig-5) So when we measure a distance, we need to know which direction we have measured. There are four main directions. East, West, North and South. There is also up and down directions. There are also direction in between the two, north-east, south-west. This way there can be many directions, really infinite number. There can be one in between any given two. Two directions which are at right angles to each other are considered separate, because movement in one direction does not affect the position in the other. In a graph, change in the X directions does not change the Y-value. Similarly, change in the Y-direction does not change the position in the X- direction. And we say X and Y are two ‘orthogonal’ directions (fig.4) In this sense our space has three dimensions because we can draw only three orthogonal directions, not more (ref. fig.6)
Defining Scalar and Vector
How do we know if a quantity is scalar (independent of direction) or vector (dependent on direction)? In simple language, if the direction is changed it must make a difference that we can experience. i) Distance: If one goes 1 km for a walk every day, will he reach the same place, every time? Of course not, it depends on the direction. So distance is a vector. ii) Force: If you apply a force to a door, will it open or close? It depends on the direction in which you apply the force. Pulling and pushing is not the same. So force is a vector. iii) Area: You are going with an open umbrella in a storm. How do you hold the umbrella? For the same wind force, the direction of the umbrella makes a difference. It may even turn inside out. The sail of a boat has to be turned in a proper direction to use the wind to best advantage, So area is also a vector. iv) Temperature, weight and Colour are not vectors. v) Volume: If you measure 5 liters it makes no difference what the direction is. Volume is a quantity that involves all three dimensions so nothing more to tell. Volume is a scalar. In conventional books, distance is called a scalar and Displacement is called a vector. There is no fundamental difference between distance and displacement. Displacement is a distance in a definite direction. But every distance has a direction. You cannot locate a town on a map given only a distance. Therefore distance is a vector. All units based on a vector will also be vectors, because if the direction is changed, it will make a difference in the other vector also. Thus area, force, velocity etc are vectors.
i) Write down if following parameters are scalar or vector. Mass, distance, velocity, volume, temperature, area, force, colour
Addition of Vector
We cannot add things which are different. How much is 5 mangos + 2 apples? It will be 5 mangoes + 2 apples. If we ask the same Questions by calling them fruit, then we can add and we have 7 fruit. Similarly, we can add 2 cms + 1 inch only if we bring them to a common unit form. 1 Inch is 2.5 cm so 2 cm + 1 inch( 2.5 cm) is 4.5cm. Vectors in different directions are like quantities in different units. You can add them only when they are in the same direction. But how do we do this? In fig 7 a force AB and AC are acting on a point A. They are vectors. They can be added only after they are made in the same form. To make them in the same form, we break up each into its components in the X and Y directions. To do this we take projections (shadows in simple parlance) on the X and Y axis. Projection of B on x axis is B” and on Y Axis it is B’. Similarly projection C on X axis is C” and on Y axis is C’. When we apply force, it may have effects in different directions. In real life, we apply forces in different direction, projection of these vectors on X and Y direction is taken to predict result of forces. Let us see an example: Two people A and B are pulling on a rope, tied to a tree. A is applying 20N force and B is applying 30N force. Both of them are applying force in 90° of each other. How much force does have and which direction? If forces A and B are applied in the directions shown, the resulting force, the sum of A and B will be as if a single force is applied in the direction marked ‘Resultant’. Plot the forces on a graph. Select proper scale. We will select 10N = 1cm on the graph. The resultant force can be measured and direction of resultant. The direction of the resultant can be seen on the graph. This method of ‘resolving’ one force (or any vector) into components in any other direction is useful in understanding how things move or behave when forces act on it.
What you have learnt
In this chapter you have learnt about basic engineering concept such work, energy, force, power. You also learnt about law of conservation of energy. You learn to calculate work done, force and power required. You also learn about scalar and vectors. You learn, how two vectors can be resolved into components using graph and carry out its addition. 2.11 Terminal Questions 1) A weight of 100kg and 5 kg drops from the height of 10m. Tell which weight reaches the ground first and why? 2) Mass of object is 20Kg, how much is the force exerted on it by earth? 3) A mass of 20kg is lifted to a height of 5m in 5 second. Calculate thefollowing a. Force (F) b. Work done (W) c. Power required (P) 4) Write down difference between scalar and vector.
ANSWER TO INTEXT QUESTIONS
2.1 i) acceleration ii) N 2.2 i) 2.3 1) work 2) rotate 3) force 2.4
Calculate your own power. Take this simple way to calculate your own power. It can be roughly calculated by measuring how quickly you can climb a stairs. 1) Select a two or three storied building. 2) Calculate the vertical height of stairs. 3) Ask your friend to measure time taken by you to climb the stairs
lot the forces on a graph. Select proper scale. We will select 10N = 1cm on the graph. The resultant force can be measured and direction of resultant. The direction of the resultant can be seen on the graph. This method of ‘resolving’ one force (or any vector) into components in any other direction is useful in understanding how things move or behave when forces act on it.
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