The conical pendulum is a fascinating mechanical system that consists of a mass attached to a string, which is fixed at one end and allowed to swing in a conical motion. This unique motion is a result of the tension in the string and the force of gravity acting on the mass. The conical pendulum is often used in physics demonstrations to illustrate concepts such as centripetal force and circular motion.
A conical pendulum is a string moving along the surface of a cone and the point object at it’s end moving along a horizontal circular motion.
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 for the conical pendulum .(time for one complete revolution).
The forces exerted on the body when it is in the position shown includes:
- it’s weight(mg)
- 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.
Describing Tensional force on conical pendulum
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 ratios:
hence:
When we make use of the relation:
The equation becomes:
This leads to:
hence we have;
dividing by sinθ on both sides;
from the equation:
therefore:
cross multiplying, the above equation becomes:
making cos θ the subject of the formular:
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
Leave a Reply