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Here we will try to solve two mathematical problems related to centripetal force. This two problems are interrelated to each other. Suppose, a particle is traveling through a circular path with uniform value of velocity. Assume that, value of velocity, v= 4 m/s, mass of particle, m= 2 kg, radius of circle, r= 2 m. Hence by calculating we get the value of centripetal force, F= 16 N, acceleration, a= 8 m/s², angular frequency, w= 2 rad/second.

Let, at t=0, particle is at point A of circle. and, at t=2, particle is at point B of circle. The angle produced by arc AB at the centre of circle is= w*t =4 rad= 229.2 degree.

Now we will determine the angle between the angle of direction of initial velocity of particle (when t=0) and the direction of final velocity (at t=2). This angle equals to the angle produced by arc AB at the centre of circle. Geometrically it can be proved easily that these two angles are equal. Let, at t=0, the direction of velocity of particle is parallel to the direction of X axis. As a result, along X axis, the value of initial velocity is 4 m/s, and along Y axis, it is 0. Since the centripetal force acts at right angle with the direction of initial velocity, along X axis the value of velocity will not change. It will change along X axis only. After 2 seconds, the value of velocity along Y axis will be 16 m/s. And at this moment the direction of resultant velocity with X axis can be determined by the following way. tan∅= (16/4)= 4, ∅= 75.96 degree. This angle is not equal to the angle produced by arc AB at the centre of circle. Please help me to find out the error. If it is possible to you, please show the correct way to solve it.