Conceptual Physics

11.2 - Sample problem: a witch and a duck balance

Since the system is stationary, it is in static equilibrium. This means there is no net torque and no net force. We use the pivot point of the scale

as the axis of rotation. We start by drawing a diagram for the problem. (The diagrams in this section are not drawn to scale; we also are not

really sure witches exist, but then we do assume objects to be massless and frictionless and so on, so who are we to complain?) We ignore the

masses of the various parts of the balance in this problem.

Draw a Diagram

Variables

Sign is important with torques: the duck’s torque is in the positive (counterclockwise) direction while the witch’s torque is in the negative

(clockwise) direction. To calculate the magnitude of each torque, we can multiply the force by the lever arm, since the two are perpendicular.

What is the strategy?

1. Draw a free-body diagram that shows the forces on the balance beam.


2. Place the axis of rotation at the location of an unknown force (the tension). This simplifies solving the problem. There is no need to
   calculate the amount of this force since a force applied at the axis of rotation does not create a torque.


3. Use the fact that there is no net torque to solve the problem. The only unknown in this equation is the weight of the witch.

Physics Principles and Equations

There is no net torque since there is no angular acceleration.

ȈIJ = 0

The weights are perpendicular to the beam, so we calculate the torques they create using

IJ = rF

A force (like tension) applied at the axis of rotation creates no torque.

The witch and the duck are balanced

on the scale. The duck weighs 44.5

N. What is the witch’s weight?

Forces xy

weight, duck 0 N ímdg = í44.5 N

weight, witch 0 N ímwg

tension 0 N T

Torques lever arm (m) torque (N·m)

weight, duck 1.65 (1.65)(44.5)

weight, witch 0.183 í(0.183)( mwg)

tension 00

(^206) Copyright 2000-2007 Kinetic Books Co. Chapter 11