Time period of conical pendulum

This article explores the time period of a conical pendulum. The time period of a conical pendulum is the time it takes for one revolution. Remember;
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Below is a diagram for a conical pendulum in motion;
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The bob on the conical pendulum above travels in the horizontal circle of radius r. The bob has mass m and is suspended by a string of length l. The tension force of the string acting on the bob is the vector T, and the bob’s weight is the vector mg. For a conical pendulum the time period (T) depends on the radius (r), angle θ between the point of suspension and the string and the strength of weight or gravitation.
In this article we shall be deriving an equation for the time period and then look at some examples involving time period of a conical pendulum. The time period (t) of pendulum is given by;
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…where l is the length of the string and θ is the angle between the string and the point of suspension of the bob or object.
Knowledge of centripetal acceleration and force will be useful here. Consider the diagram abovel there are two forces acting on the bob;

• the tension T in the string, which is exerted along the line of the string acts toward the point of suspension
• the downward bob weight mg, where m is the mass of the bob and g is the gravitational acceleration.

The force exerted by the string can be resolved into a horizontal component, Tsinθ, toward the center of the circle, and a vertical component Tcosθ in the upward direction. 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.
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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;
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We can form am expression for T in both equations;
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…we can now compare both expressions to get;
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…we can get rid of m on both sides;
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Since the speed of the pendulum bob is constant, it can be expressed as the circumference 2πr divided by time t required for one revolution of the bob;
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We can substitute v in the equation above to get;
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We can rearrange and use the trigonometric identity tanθ = sinθ / cosθ to make t the subject;
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The radius is very hard to measure which you probably will never record in an experiment so we can replace r = lsinθ. This can be read from the triangle;
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…therefore;
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