Conical pendulum

A conical pendulum consists of a weight (or bob) fixed on the end of a string or rod suspended from a pivot.

The conical pendulum was first studied by the English scientist Robert Hooke around 1660[1] as a model for the orbital motion of planets.

[2] In 1673 Dutch scientist Christiaan Huygens calculated its period, using his new concept of centrifugal force in his book Horologium Oscillatorium.

Consider a conical pendulum consisting of a bob of mass m revolving without friction in a circle at a constant speed v on a string of length L at an angle of θ from the vertical.

From Newton's second law, the horizontal component of the tension in the string gives the bob a centripetal acceleration toward the center of the circle: Since there is no acceleration in the vertical direction, the vertical component of the tension in the string is equal and opposite to the weight of the bob: These two equations can be solved for T/m and equated, thereby eliminating T and m and yielding the centripetal acceleration: A little rearrangement gives: Since the speed of the pendulum bob is constant, it can be expressed as the circumference 2πr divided by the time t required for one revolution of the bob: Substituting the right side of this equation for v in the previous equation, we find: Using the trigonometric identity tan(θ) = sin(θ) / cos(θ) and solving for t, the time required for the bob to travel one revolution is In a practical experiment, r varies and is not as easy to measure as the constant string length L. r can be eliminated from the equation by noting that r, h, and L form a right triangle, with θ being the angle between the leg h and the hypotenuse L (see diagram).