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Dynamics of Circular Motion ౼ Problems and Solutions

1. A small block of mass 100g moves with constant speed in a horizontal smooth circular groove, with vertical side walls, of diameter 50 cm. If the block takes 2s to complete one round, find out the normal contact force by the side wall of the groove.

The block executes a uniform circular motion in a horizontal circular groove, with vertical side walls. The only source of required centripetal force for that circular motion to take place is the vertical side walls of the groove (as the friction is absent). The normal contact force applied by the vertical walls of groove on the block is always directed towards the centre of the circular groove and it is only this force which is responsible for setting the particle in uniform circular motion. 

Also the weight of the block is balanced by the other normal contact force arising from the horizontal base surface of the groove. This balancing of vertical forces keeps the block in a fixed horizontal plane.

Applying Newton's second law of motion along radial direction, we can write
Dynamics of Circular Motion
where Fₙ and aₙ denote respectively the normal force (or radial force) and normal acceleration. The problem has asked to find this Fₙ. To do this, we first need to have the value of speed v of the block. It is for this purpose the time period of the circular motion has already been provided in the problem.
Circular Motion Problems
Therefore the vertical wall of the groove constantly applies a radial force of magnitude 0.25 N on the block.

2. A particle of mass m moves in a circle of radius R with a constant speed v. Find the average force on the particle, after covering a half-circle i.e. a semi-circle.

By definition of acceleration, we can write
Dynamics of Circular Motion
Dynamics of Circular Motion
Considering the diagram shown above, we have:
      Initial Velocity = +v
      Final Velocity  = -v
(You should note that the + and - signs are used here just to indicate or differentiate the two opposite directions of the velocity vector.)
Change in Velocity = Final Velocity - Initial Velocity

or, Δv = (-v) - (+v)
or, magnitude of change in velocity |Δv| = 2v