Energy and Thermal Physics

The automatically straight-line graph

Teaching Guidance for 14-16 PRACTICAL PHYISCS

Examples of using a straight line graph to find a formula.

Example 1: To show that πR 2 gives the area of a circle.

For any circle π is the number 3.14 in the equation:

circumference = 2π x radius or π x diameter

So π is circumferencediameter

Starting from that (as a definition of π) we can show that the area of a circle is πR 2.

Draw a large circle with centre 0 and radius R. Plot a graph of 2πr upwards against r along.

Then the graph must be a straight line and its slope will be 2π.

The end-point A, of the graph belongs to a big circle of radius R. Each other point of the graph: line 0A belongs to a smaller circle, of radius r.

Sketch III shows two small circles close together with radii (r) and (r + tiny bit of radius).

What is the area of the shaded ring between them? The ring has width (tiny bit of radius) and lengthr (its circumference). Its area is 2πr x (tiny bit of radius).

On the Graph IV the shaded pillar shows just that same area, 2πr x (tiny bit of radius).

Now ask about all such rings from the centre 0 out to radius R. Their total area is the same as the area of all the pillars in Graph V. That is the triangle of height2πR and base R.

AREA = ½2πR x R = πR 2.

Therefore area of circle is πR 2.

Example 2: To show that s = ½at 2 for constant acceleration from rest.

Plot a graph of at upwards against t along. Then with a constant the graph must be a straight line; and its slope will be a (Graph VI).

Choose a tiny bit of time on the t-axis and draw a pillar up to the line (Graph VII). The area of the pillar is: height x width,

(at) x (tiny bit of time)

and that is (v) x (tiny bit of time), since acceleration x time is speed.

And that is (tiny bit of time travelled).

Then total distance travelled, s, is given by the total area of all such pillars (Graph VIII).

s = area of triangle OAB = ½at x t = ½ at 2.

appears in the relation F=ma a=dv/dt a=-(w^2)x
is used in analyses relating to Terminal Velocity
can be represented by Motion Graphs

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