Conical Pendulum
A conical pendulum is similar to the simple pendulum except the motion also involves a circular motion about the point of suspension thus creating a conical shape. The conical pendulum was first studied by the English scientist Robert Hooke in 1660 as a model for orbital motion of the planets. Later in the 1800s conical pendulums were used as the timekeeping element in clockwork timing mechanisms where a smooth motion was required. This article explores of a conical pendulum. We shall explore quantities such as forces/centripetal forces, velocity, time period, and how these quantities relate to each other in a conical pendulum.
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Consider the conical pendulum consisting of mass (m) revolving without friction in a circle at constant speed v on a string of length L at an angle of θ from the vertical. The figure below should help analyse properties of a conical pendulum.
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In the diagram above notice there are two forces acting on the bob;
- There is the tension (T) in the string which is exerted parallel to the string and acts toward the point of suspension.
- The bob has weight mg acting downward. …where m is the mass of the bob and g is the gravitational acceleration.
Knowledge of centripetal force and acceleration will be useful here. Circular motion is caused by a centripetal force. The magnitude of the centripetal tangent speed v along a path with radius of r is given by;
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…where ac is the centripetal acceleration. The direction of the force is toward the centre of the circle about which the object is moving or oscillating. When you look at the conical pendulum from above you observe a circular path of the object as shown below.
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Here is a conical pendulum again from above;
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We can resolve the forces acting on the bob in a vertical component or a horizontal component. The bob or mass on the string is only moving in the horizontal plane which means the resultant force in the vertical plane sums up to zero; Below is the force diagram;
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The tension/force in the string is acting at an angle θ. That must mean;
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Since there is no motion in the vertical plane that must mean that;
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So when the bob on the string is rotating in equilibrium along the vertical plane;
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I we look at the force diagram trigonometrically there are other properties to notice such as;
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In circular motion you should know that;
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Considering the fundamentals above we can find other unknown quantities about the motion in a conical pendulum. For example in an experiment you may want to find the length of the string, velocity of the bob or object, height from the plane of revolution to the point of suspension, time period (time for one revolution), frequency (number of revolutions per second)
We shall analyse some of the quantities below.
Analysing time period
In this section we shall analyse the time period of a conical pendulum using the fundamentals explored above; We already know that;
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…and 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, therefore;
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In both equations we can make T the subject where;
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…now we can compare them;
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…we can cancel out m on both sides to get;
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Here we’re going to represent time period with the letter (t) so;
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…we know that;
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…so we can replace v in the equation above to get;
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Now make t the subject to find the time required for the bob to travel one revolution. We can use the trigonometric identity tanθ = sinθ/cosθ which will give;
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In a practical experiment it would be very had to measure the values of radius if they vary. We can eliminate r from the equation by noting from the diagram;
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So substituting the value of r in the equation gives;
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We can simplify it further to get;
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Analysing height of suspension from pendulum
In this section we shall come up an equation to find the height (h)
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We already know that;
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cosθ and sinθ can make tanθ, so;
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If you look at the diagram above you can see that;
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…that must mean that;
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[EXAMPLES & EXCIRCISES MISSING]
