UNIFORM CIRCULAR MOTION

 

Uniform circular motion:  It is a motion of an object traveling along a circular arc or a circle with constant speed.  Radius of the circle and speed of the object are constant.

Centripetal acceleration:  Centripetal means centre-seeking.  Centripetal acceleration is the instantaneous acceleration directed toward the centre of the curvature.  Magnitude of centripetal acceleration is constant, but its direction changes, which is always directed toward the centre of the circle.

  

ac: Centripetal acceleration (m/s2): 
v:  Speed of the object undergoing uniform circular motion (m/s).  this is a constant speed
r:  Radius of the curvature of the path (m)
rpm: Revolutions per minute
T:  Period (second, s)
f:  Frequency, number of revolution per second (1/s, Hertz, Hz)
m:  Mass of the object (kg)
Fc:  Centripetal Force (Newtons, N)

Centripetal Force (Newtons, N):  Centripetal Force is the net force (vector sum of all forces) that is always directed perpendicular to the velocity of the body and towards the instantaneous center of curvature of the path.  Centripetal force is also always perpendicular to the axis of the rotation.  This force is created by centripetal acceleration.  F = m x a

 

 

Centrifugal Force (Newtons, N):

As an object travels in a circle, it appears to have a force pulling it outward.  This is called a centrifugal force.  It is also called as fictitious force, inertial force or non existence force.  Centripetal and centrifugal forces are equal in magnitude, but opposite in direction.  centrifugal forces are directed away from the centre of the curvature.

 

EXAMPLE 1:

A ball is spinning in a vertical circle at the end of a string that is 0.85 m long. The ball has a mass of 2.04 kg and moves at a constant speed of 5 m/s.  Determine the tension in the string when the ball is at the top of the circle and at the bottom of the circle

Fg:  force of gravity on the ball (N)

FT:  tension in the string (N)

Fc:  Centripetal force (N)

Centripetal Force is the net force (vector sum of all forces):  Fc = Fnet

Fc = FT + Fg

FT = Fc - Fg

 

Fc = (2.04 kg).(5 m/s)2/(0.85 m) = 60 N

Fg = m.g = (2.04 kg).(9.8 m/s2) = 20 N

 

Tension at the top of the circle:

FT = Fc - Fg

Let assume: (up is positive) and (down is negative)

FT = (-60 N) - (-20 N) = - 40 N

Tension at the top of the circle:  FT = 40 N [down]

 

Tension at the bottom of the circle:

FT = Fc - Fg

FT = (+60 N) - (-20 N) =  80 N

Tension at the bottom of the circle:  FT = 80 N [up]