Horizontal Circular Motion

This lesson covers important applications of circular motion in the horizontal plane: the conical pendulum, banked curves, and roundabouts.

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1 The Conical Pendulum

A particle of mass m is attached to a string of length l. The string makes angle theta with the vertical as the particle moves in a horizontal circle.

Setup

The radius of the circle is: r = l sin theta

The forces on the particle:

• Weight mg downward
• Tension T along the string towards the point of support

Resolving Forces

Vertically (no vertical acceleration): T cos theta = mg ... (1)

Horizontally (centripetal): T sin theta = mv^2/r = mr omega^2 ... (2)

Dividing (2) by (1):

tan theta = v^2/(rg) = r omega^2/g

Since r = l sin theta: tan theta = l sin theta * omega^2/g

So: omega^2 = g/(l cos theta)

And the period: T = 2pi sqrt(l cos theta / g)

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2 Worked Example 1

A conical pendulum has string length 0.8 m. The string makes 30 degrees with the vertical. Find the angular speed and the period.

Solution: