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Question 1: What are the centripetal acceleration and centripetal force? Derive their equations.

ANSWER

We know that when no unbalanced force is acting on a moving body, it will continue to move in a straight line with uniform velocity. A body moving in a circle or curve, therefore, is experiencing some force compelling it to move in that path.

Centripetal Force

The force, which compels a body to move in a circular path, is called centripetal (or center-seeking) force.

Mathematically,

centripetal force equation                                                                                                            …           (1)

Centripetal Acceleration

The change in the direction of the velocity of the body produces acceleration that is directed to the center of the circle, called centripetal (or center-seeking) acceleration, denoted by vecac.

Mathematically,

equation centripetal accelaration                                                                                                      …              (2)

Equation for centripetal accelerationfig cent acceleration

Consider a body of mass m is moving with uniform velocity v in a circle of radius r and center at C. At point ‘A’ at time t1, velocity of the body is vecv . Then at point ‘B’ when time is t2, its velocity changes to vecvdash .  Now let us construct a triangle A’B’C’ so that A’B’ is equal and parallel to vecv  and A’C’ is equal and parallel to vecvdash .  As the speed does not change, the velocity has the same magnitude with a changed direction. The change in velocity is delv in a time interval Δt = t2– t1.

When Δt is very small, Δvecv  is also very small and the arc arcABis approximately equal to cord AB. Then on comparison, ΔACB and ΔA’B’C’ are isosceles. So,

isoseles

If Δt is very small, and hence do θ, point B is very close to point A, then,

S = rΔt

Use this value of ’s’ in the above equation.

ch5para1

The RHS is the instantaneous acceleration. Thus

ch5para2

This is the required equation of centripetal acceleration.
In vector form,

ch5para3

Here vaca  is the centripetal acceleration.
unitvecr is the unit vector along the radius acting outward from the center of the circle.

Both the centripetal acceleration and the radius vector are, therefore, opposite in direction with each other. Therefore, we have,

centripetal acceleration vector

This is the equation of centripetal acceleration.

Equation for centripetal force

From Newton’s Second Law of Motion, F = ma

Putting the value of acceleration, we get,

ch5para4

This is the required equation for the centripetal force.

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