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:

omega^2 = g/(l cos theta) = 9.8/(0.8 cos 30) = 9.8/(0.8 * sqrt(3)/2) = 9.8/0.6928 = 14.14

omega = 3.76 rad s^(-1)

T = 2pi/omega = 2pi/3.76 = 1.67 s

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3 Worked Example 2

A conical pendulum has a string of length 1.2 m and the bob has mass 0.5 kg. It revolves at 2 revolutions per second. Find the angle the string makes with the vertical and the tension.

Solution:

omega = 2pi * 2 = 4pi rad s^(-1)

omega^2 = g/(l cos theta), so cos theta = g/(l omega^2) = 9.8/(1.2 * 16pi^2) = 9.8/189.47 = 0.0517

theta = arccos(0.0517) = 87.0 degrees (nearly horizontal!)

T = mg/cos theta = 0.5 * 9.8/0.0517 = 94.8 N

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4 Banked Curves

When a car goes around a banked curve (angled road surface), the banking angle helps provide the centripetal force, reducing the need for friction.

No Friction Case

On a frictionless banked road at angle theta:

Resolving vertically: N cos theta = mg

Resolving horizontally: N sin theta = mv^2/r

Dividing: tan theta = v^2/(rg)

This gives the "design speed" for the bank -- the speed at which no friction is needed.

Worked Example 3