The diagram below represents a body of mass m moving on a horizontal circle with a constant velocity v at the edge of a cord of length L.
As the body swings around it’s path the cord swings over the surface of a cord. The cord makes an angle θ with the vertical so that the radius of the circle in which the moves is R=Lsinθ and the magnitude of the velocity is given by:
where T is the periodic time.(time for one complete revolution).
The forces exerted on the body when it is in the position shown are it’s weight(mg) and tension(T) in the cord.
Tension T can be resolved into horizontal and vertical component Tsinθ and Tcosθ respectively.The body has no vertical acceleration, so the vertical forces Tcosθ and mg are equal in magnitude.
The resultant inward radial(centripetal) force is the horizontal component Tsinθ which is equal to the mass(m) multiplied by the radial or centripetal acceleration.
From linear motion, F=ma.
but since the motion is circular; a=v2/r
hence, the expression for the horizontal motion.
For the vertical forces;
Tsinθ = mg ……………………………….. (ii)
When the equation (i) is divided by equation (ii) the result is:
but from trigonometry, sinθ/cosθ = tanθ; hence
When we make use of the relation r = L sinθ and v = 2πLsinθ/T
The equation becomes:
This leads to:
hence we have;
dividing by sinθ on both sides;
from the equation tanθ = sinθ/cosθ ; 1/cosθ = tanθ/sinθ and therefore:
and hence;
and making T the subject of the formular, we have;
which will results to:
The angle θ depends on the time of revolution T and the length of the cord.
For a given length L, cosθ decreases as the time is made shorter and shorter and the angle θ increases. The angle θ never becomes 90o sinces this requires that T = 0 or v=∞ .
The largest possible value of T is given by T=2π√l/g which occurs for a very small angle θ for which cosθ=1 and therefore:
Related topics
References
- Sanny, J. S., & Ling, S. (2016). University Physics Volume 1. OpenStax. https://doi.org/13: 9781938168161
- Abbott, A. F. (1989). PHYSICS (5th ed.). Heinemann. https://doi.org/978043567014