Size+of+the+Circle+&+Speed+(Math)

Earlier, by considering the examples we recognized that when an object is moving along a circle the spped at which the movement can be made is proportional to the radius of the circle.

Let us see if this recognition is true in terms of physics.

Assuming that the speed during the turn is constant, the speed can be determined using the simple relationship.



If an object traversed the full length of a circle of radius R, the distance travelled is equal to the circumference length 2πR . Let the time taken for this cyclic journey be T.


 * Note that T is a constant, as no matter how many turns the object is making, if the object is moving at a constant speed the time taken for a single cycle is always same.
 * Also, note that if the distance travelled along the circular path is a fraction of a circle, then the numerator can be multiplied by that factor to indicate the correct distance, but then the denominator also to be multiplied by the same factor to indicate the correct time. The same factor in numerator and denominator cancel off yielding the same relationship given by equation (01)

For example, if a circular turn only spanned a quarter of a circle then the distance travelled will be ¼ of the circumference length (2π R) and the time taken for the turn will be ¼ of T;



Now the above relationship can be re-written as follows;



All the terms inside the brackets are constants. Therefore we can write the term inside the bracket as a single constant ‘k’



The constant ‘k’ can be replaced with the proportionality symbol (α).



i.e. The above result which we derived mathematically according to the rules of physics is justifying the behavior we recognize considering the examples. This relationship can be rephrased as follows.