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Moments and Torque
Torque, also called moment or moment of force, is the tendency of a force to
rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull,
a torque can be thought of as a twist.
d1
d2
F1
F2
Unfortunately for the donkey, the moment of the
donkey about the axle (𝐹1 ∙ 𝑑1 ) is less than the
moment of the cargo about the axle (𝐹2 ∙ 𝑑2 ).
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Definition of a Moment
d
F
𝑀 = 𝑚𝑜𝑚𝑒𝑛𝑡 = 𝐹 ∙ 𝑑
A
UNITS:
𝑓𝑜𝑟𝑐𝑒 ∙ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑁∙𝑚
𝑖𝑛 ∙ 𝑙𝑏𝑠
𝑓𝑡 ∙ 𝑙𝑏𝑠
The moment or torque of the force F about point A is a measure of the tendency
of F to produce rotation about point A.
Notice that d is the shortest distance from the line of action of F to point A. In
other words, 𝑭 ⊥ 𝒅.
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Consider the moment of F about point A for the cases below.
line of action
d
A
A
𝑀 =𝐹∙𝑑
A
d
F
𝑀 =𝐹∙𝑑
d
F
𝑀 =𝐹∙𝑑
A
F
F
𝑀 =𝐹∙𝑑 =𝐹∙0=0
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Class Example
The semicircular segment pivots about point A.
a. What is the moment of F1 about A?
b. What is the moment of F2 about A?
𝐹 = 100 𝑙𝑏𝑠
𝑑 = 14 𝑖𝑛
a. 𝑀 = 𝐹1 ∙ 𝑑
𝑀 = 100 𝑙𝑏𝑠 ∙ 14 𝑖𝑛 = 1,400 𝑖𝑛 ∙ 𝑙𝑏𝑠
b. 𝑀 = 𝐹2 ∙ 𝑑
A
𝑀 = 80 𝑙𝑏𝑠 ∙ 0 𝑖𝑛 = 0
F2 = 80 lbs
F1 = 100 lbs
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Class Example
Find the moment of F about point A.
d1 = 2m
𝐹𝑥 = 100𝑁 ∙ cos 55° = 57.4𝑁
A
𝐹𝑦 = 100𝑁 ∙ 𝑠𝑖𝑛 55° = 81.9𝑁
d2 = 3m
+ 𝑀𝐴 = −𝐹𝑥 ∙ 𝑑2 + 𝐹𝑦 ∙ 𝑑1
= −57.4𝑁 ∙ 3𝑚 + 81.9𝑁 ∙ 2𝑚
= −172.2 𝑁 ∙ 𝑚 + 163.8 𝑁 ∙ 𝑚
𝑀𝐴 = −8.4 𝑁 ∙ 𝑚 = 8.4 𝑁 ∙ 𝑚
= 8.4𝑁 ∙ 𝑚 𝐶𝐶𝑊
55°
F = 100N
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What is the difference between a moment and a torque?
The words moment and torque are often used interchangeably. A moment and a
torque are physically the same thing; they are both a measure of the tendency of a
force or set of forces to produce rotation.
Moment is a more general term than torque. A torque is a special application of a
moment and generally refers to the tendency to produce rotation about an axle.
Typical usage of these words can vary by discipline and location.
•
•
A gearmotor can output a peak torque of 6N-m (physical rotation occurs about the
motor shaft).
The weight of a child who climbs out on the branch of a tree will induce a moment on
the limb; the moment will be maximum at the base of the branch.
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Class Problem
Uncle Fred who weighs 200 lbs. is riding a Ferris wheel at the county fair along with his 80 lb. son, Freddie.
The Ferris wheel has a diameter of 30 feet.
a. What moment will Uncle Fred exert about the center of rotation (point A) when q = 45° if he is the only
one riding? Will this moment act clockwise (CW) or counter clockwise (CCW)?
b. What moment will Freddie exert about the center of rotation (point A) when q = 45° if he is the only
one riding (his angle is actually 45° + 90° = 135°)? Will this moment act CW or CCW?
c. What is the net moment about point A when q = 45° and both are riding? Assume that a CW moment is
positive and a CCW moment is negative. Is the net moment CW or CCW?
d. Derive an equation for the moment about point A as a function of q when both people are riding.
e. Using Mathcad, plot the net moment for q = 0° to q = 360°.
Assumptions:
• Ignore dynamic effects.
• Assume the weights of Uncle Fred and Freddie
are centered at the pin where their chairs are
connected to the circular structure.
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