Consider an experiment where a stopper is attached to a string which is also attached to a hanging mass at the other end. The radius of circular motion is constant (for the time-being), and the stopper is swung in a horizontal, circular path as a result of the gravitational acting on the hanging mass which causes tension throughout the string which provides the force for the centripetal acceleration of the string. The period of revolution is measured for different radii of motion tested. What I want to know is what is the relationship between the period and radius? I'm confused because v=2Pi r/T which would suggest a directly proportional relationship, but Centripetal force = 4(Pi)^2 mr/T^2 which would suggest otherwise... Can someone please clarify this relationship for me?
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The tension in the string, which is a constant, provides the centripetal force but also one component of it provides a force to support the stopper against gravity. However if the rate of spin is high and the weight of the stopper is small we can neglect the weight of the stopper and the centripetal force will be the tension in the string. So the centripetal force is constant for all trials ( the weight of the hanging mass ).
F(centripetal) = m v^2 /r
But as you said v = 2 pi r /T
Substituting gives
F = 4 m pi^2 r / T^6
F, m, 4 and pi are constants, so
r / T^2 = k, or
r = k T^2
r is proportional to T^2
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The reason why you could not use v = 2 pi r /T to find the relationship of T to r is that v is also a variable.
F(centripetal) = m v^2 /r
But as you said v = 2 pi r /T
Substituting gives
F = 4 m pi^2 r / T^6
F, m, 4 and pi are constants, so
r / T^2 = k, or
r = k T^2
r is proportional to T^2
.
The reason why you could not use v = 2 pi r /T to find the relationship of T to r is that v is also a variable.
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to calculate the speed = V, of a particle moving in a circular path of radius = R,
at constant speed: U take the length of the path = 2πR and divide by the time
of one revolution = T. But this is just the definition of speed (or avg speed) and
works for circular as well as linear constant speed motion.
In constant speed circular motion, in particular, we know that the size of the
acceleration = a = V²/R. and by definition V = path length/T = 2πR/T, so
a = constant = (2πR)²/T²
aT² = 4π²R
so the relationship between the period T and the radius R for constant speed
circular motion is:
T is proportional to √R
or
R is proportional to T²
at constant speed: U take the length of the path = 2πR and divide by the time
of one revolution = T. But this is just the definition of speed (or avg speed) and
works for circular as well as linear constant speed motion.
In constant speed circular motion, in particular, we know that the size of the
acceleration = a = V²/R. and by definition V = path length/T = 2πR/T, so
a = constant = (2πR)²/T²
aT² = 4π²R
so the relationship between the period T and the radius R for constant speed
circular motion is:
T is proportional to √R
or
R is proportional to T²