A conical pendulum is a pendulum that consist of a mass hanging from a string of length that describes a circle when in motion, see diagram on the left . The characteristic of this motion can be obtained in terms of the length of the string and the angle with respect to the vertical. In order to start solving this motion, the first selection is of a suitable frame of reference; in this case, a xy-coordinate system that travels with the mass is the most appropriate, . In this drawing, the radius of the circle is also shown. This radius can be calculated from the triangle formed by the string, the vertical, and the radius of the circle, , resulting in . The next step is to identified the forces acting on the mass, in this case, there are two forces acting on the mass, the weight of the object, , and the tension of the string, . In order to complete the vector addition of this vector, it is convenient to express the tension of the string in terms of the components along the x and y axis, . The values of these components are calculated after the angle has been recognized in the lower triangle of the drawing (equivalent to the larger triangle, ). Thus, the components are and .The last drawing of the sequence includes the velocity of the mass. The velocity is perpendicular to the plane form by the x and y axis, .

To solve the remaining part of this setting the standard procedure involving Newton's laws of motion is applied.

y-component: with the acceleration in the y-direction being zero, . Newton's second law for this component is from where the tension of the string may be found, .

x-component: In this case, there is only one force in this direction, with the acceleration in the x-direction being the centripetal acceleration, . Thus, applying Newton's second law to this component, the following equation is established,  . Substituting the values of the centripetal acceleration and the tension of the string, . Substituting in this relation the value of the radius, , the equation becomes, . From this last expression, if the angle is given, the value of the velocity may be obtained,

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com