Although simple architectural structures like bridges, catwalks, cantilevers, and Stonehenge have been integral in human societies for millennia, as have levers and other simple tools, modern students of introductory physics continue to grapple with Newton's conditions for static equilibrium. As formulated in typical introductory physics textbooks,1–4 these two conditions appear as
(1)
and
(2)
where each torque τ is defined as the cross product between the lever arm vector r and the corresponding applied force F,
(3)
having magnitude,
(4)
The angle θ here is between the two vectors F and r. In Eq. (1), upward (downward) forces are considered positive (negative). In Eq. (2), counterclockwise (clockwise) torques are considered positive (negative). Equation (1) holds that the vector sum of the external forces acting on an object must be zero to prevent linear accelerations; Eq. (2) states that the vector sum of torques due to external forces about any axis must be zero to prevent angular accelerations. In our view these conditions can be problematic for students because a) the equations contain the unfamiliar summation notation Σ, b) students are uncertain of the role of torques in causing rotations, and c) it is not clear why the sum of torques is zero regardless of the choice of axis. Gianino5 describes an experiment using MBL and a force sensor to convey the meaning of torque as applied to a rigid-body lever system without exploring quantitative aspects of the conditions for static equilibrium.
1.
Raymond A.
Serway
and
Chris
Vuille
,
College Physics
, 9th ed. (
Brooks/Cole Cengage Learning
,
2012
), pp.
240
247
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Raymond A.
Serway
and
John W.
Jewett
Jr.
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Physics for Scientists and Engineers
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Brooks/Cole Cengage Learning
,
2010
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349
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and
Gene
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Nicholas J.
Giordano
,
College Physics: Reasoning & Relationships
, 2nd ed. (
Brooks/Cole Cengage Learning
,
2013
), p.
255
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5.
Concetto
Gianino
, “
An MBL experiment to analyze the torque on a rigid body
,”
Phys. Teach.
47
,
224
(
April 2009
).
6.
Vernier Software & Technology, www.vernier.com.
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Raul T.
Mainardi
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Demonstration experiments with platform scales
,”
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39
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488
(
Nov. 2001
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G.
Yoder
and
J.
Cook
, “
Innovative interactive lecture demonstrations using wireless force sensors and accelerometers for introductory physics courses
,”
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567
(
Dec. 2010
).
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Mario
Belloni
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A simple demonstration for the static ladder problem
,”
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(
Nov. 2008
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11.
Model Human Arm (PS-2611), www.pasco.com.