Let us do a very, very interesting sum, the sum is.
If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body. Express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled by the body is 2 units, 0 unit. Graphical part later, first let’s form the equation. Very interesting sum, let’s go step by step. What is given, what we are supposed to find?
An equation with two variables. For that some condition is given. What is the condition given? The condition given is the work done by a body is directly proportional to the distance. Work done and distance are the two quantities. Work done and distance, assume one as x and assume one as y to form the equation, right.
So, let us assume x as the distance and y as the work done, okay. You can assume x as work done, y as distance. Choice is yours. So, let x be the distance and y be the work done. Now, what is the condition given, work done that is y, is directly proportional to the distance that x. What is the relation that we get? Y is directly proportional to x. Variation y is directly proportional x, to convert this into an equation we need to introduce a constant of variation. So, let us write this as y is equal to kx, where say K is the constant of variation. Is this thing clear? But what is the constant of variation given to us. The constant of force is given as 5 units. So, we come to a conclusion that K is equal to 5 because K is that constant. So, what is the equation that we will get? Y is equal to kx, K is a constant and k is given as 5. So, we come to a conclusion that the equation will be y is equal to 5x.
The only trigger that you get here is once you assume one as x and other as y. Otherwise, when you read the sum and only keep on trying to understand it, you will never be able to form these equations. So, the first thing you do is assume one quantity as x, the other as y. Then it is given directly proportional, so you directly write y as directly proportional to x. Introduce a constant that is K, so y is equal to kx. K is 5, so y is equal to 5x, as simple as that.
Now, let’s form the graph, let’s draw the graph of this. To draw the graph we need to form the table with x, y, (x,y). Now assume the value of that variable which is there in the right hand side. We have got x in the right hand side. Any three values of x. Say x is 0, 1 and minus 1 or 2, whatever is fine. Now, let’s substitute the values of x to find the values of y. So, when x is equal to zero, we get y is equal to 5 into zero. We get y is equal to zero. The point is 0, 0. When x is equal to 1, you get y is equal to 5 into 1. You get y is equal to 5. You get the point as 1, 5. When x is equal to minus 1, y is equal to 5 into minus 1, y is equal to minus 5. So, the point is minus 1, minus 5.
Now, let’s plot these points on the graph paper. No need to show the calculations, right. Let’s do it. The scale, the normal scale 1 cm is equal to 1 unit. The first point is 0, 0 that is the origin. The second point is 1, 5 in the first quadrant and the third is minus 1, minus 5 which will be in the third quadrant. First quadrant 1, 5, origin 0, 0 and minus 1, minus 5 is the third quadrant, right.
Now, you can see that all the points are collinear. So, you can always draw a straight line passing through them. Now, interesting what does x represent, the distance, what does y represent, the work done? X is the distance, y is the work done. So, all the x coordinates represent the distance. Y coordinates represent the work done, right. Also read from the graph the work done, the work done that means you are suppose to find y. When the distance travelled by the body is 2 units and 0 units. When the distance is 2 and 0. Distance is 2 and 0, so, can we say x is 2 and 0 because x is the distance. So, x is 2, x is 0. You are suppose to find y. You can directly substitute in the equation but then here it is given from the graph. So, from the graph how do we do it? Let’s see, after you write the equation, let’s see how you do it from the graph. The distance is 2 and the distance is 0. So, for the distance travelled 2 units draw a line perpendicular from there, right. So, what do we get, right, from 2 you take to that line and from there you put it to 10, you get work done as 10 units, clear. Similarly for the distance travelled zero, the work done is also zero. So, the work done is also zero units. Hope you understood? From 2 you took a perpendicular to the line and from there on the y axis, from 2 on the x axis because the distance is 2 to the line from that point to y again a perpendicular wherever it intersects 10 units, is the answer for the first one. And second distance is zero, work is zero. That goes without saying, right, very, very simple.
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