Forging new generations of engineers. Moments A lesson on understanding and evaluating moments of...

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Forging new generations of engineers

Moments

A lesson on understanding and evaluating

moments of forces

In This Lesson

• What is a moment?• How are moments calculated?• How are moments evaluated in dynamics

problems?• How are moments evaluated in static

equilibrium problems?• How do moments affect unconstrained

objects?

What Is a Moment?

The moment or torque of a force is a measure of the tendency of the force to rotate the body upon which it acts about an axis.

Terminology

pivot distance

lever arm

Formula for Moment

MomentMM = F x D

distance= D

Units for Moments

Force Distance Moment

English

CustomaryPound (lb) Foot (ft) lb-ft

SI Newton (N) Meter (m) N-m

The Right Hand Rule

Right Hand Rule

POINTS

TOWARD

YOUPOSITIVE

Right Hand RuleTHUMB POINTS AWAY FROM YOU

NEGATIVE

Moment Calculations

WrenchF = 20 lb

D = 9 in.

M = -(F x D)

***Use the right hand rule to determine positive and negative.

D = 9 in. = .75 ft

M = -(20 lb x .75 ft)

M = -15 lb-ft

(15 lb-ft clockwise)

Moment Calculations

Longer WrenchF = 20 lb

D = 1 ft

M = -(F x D)

M = -(20 lb x 1 ft)

M = -20 lb-ft

Moment Calculations

L-shaped Wrench

F = 20 lb

D = 3 in. = .25 ft

M = -(F x D)

M = -(20 lb x .25 ft)

M = -5 lb-ft¯

Moment Calculations

Offset WrenchF = 20 lb

D = 8 in. + 10 in. = 1.5 ft

M = -(F x D)

M = -(20 lb x 1.5 ft)

M = -30 lb-ft¯

10 in.

D = r = 50 cm = .5 m

M = F x D

M = 100 N x .5 m

M = 50 N-m

Moment Calculations

Wheel and Axle

F = 100 N

r = 50 cm

Fy = Fsin50° = (100 N)(.766)

Fy = 76.6 N

D = r = 50 cm = .5 m

M = Fy x D

M = 76.6 N x .5 m

M = 38.3 N-m

Moment Calculations

Wheel and Axle

F = 100 N

r = 50 cm

What Is Equilibrium?

The state of a body or physical system at rest or in unaccelerated motion in which the resultant of all forces acting on it is zero. The sum of all moments about any point or axis is zero.

ΣM = 0

M1 + M2 + M3 . . . = 0

Moment Calculations

See-Saw

ΣM = 0

M1 + (–M2) = 0

M1 = M2

F1 x D1 = F2 x D2

25 lb x 4 ft = 40 lb x D2

100 lb-ft = 40 lb x D2

Moment Calculations

See-Saw

F1 = 25 lb

F2 = 40 lb

D1 = 4 ft D2 = ? ft

40 lb 40 lb

2.5 ft = D2

ΣM = 0

MB + (–MC) = 0

MB = MC

RB x DAB = FC x DAC

RB x 10 ft = 35 lb x 3 ft

RB x 10 ft = 105 lb-ft

Moment Calculations

Loaded Beam

DAB = 10 ft

DAC= 3 ft

FC = 35 lbRB

10 ft 10 ft

RB = 10.5 lb

RA + RB = 35 lb

RA = 35 lb – 10.5 lb = 24.5 lb

Select the pivot location A. Solve for RB.

Fc = 600 lb

Moment Calculations

24 ft 8 ft

FB = 500 lb

Replace the pinned and rolling supports with reaction forces.

DAC = 24 ft

DCD = 8 ft

DCB = 12 ft

DAD = 32 ft

Fc = 600 lb

Moment Calculations

DAC = 24 ft

DCD = 8 ft

DCB = 12 ft

DAD = 32 ft

24 ft 8 ft

FB = 500 lb

Select the pivot at A. Solve for RDY.

ΣM = 0

MD – MB – MC = 0

MD = MB + MC

RDY x DAD = (FB x DCB) + (FC x DAC)

RDY x 32 ft = (500 lb x 12 ft) + (600 lb x 24 ft)

RDY x 32 ft = 6000 lb-ft + 14400 lb-ft

RDY x 32 ft = 20400 lb-ft 32 ft 32 ft

RDY = 637.5 lb

Moments on An Airplane

AILERON Roll

ELEVATORS Pitch

RUDDER Yaw

PATH WITH MOMENTS

Translation and Rotation

PATH WITHOUT MOMENTS

Pure Translation

Moments on an Airplane

References

Halpern, A.M. (1988). Schaum’s 3000 solved problems in physics. New York, NY: McGraw-Hill.

NASA. (n.d.). The beginners' guide to aeronautics. Retrieved

June 11, 2008, from http://www.grc.nasa.gov/WWW/K-12/airplane/

Nave, C.R. (2005). HyperPhysics. Retrieved June 12, 2008,

from http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html

National Institute of Standards and Technology. (2000). The NIST reference on constants, units and uncertainty. Retrieved June 11, 2008, from http://physics.nist.gov/cuu/Units/

Writer: Wendy DeMane

Content Editor: Wes Terrell

Production Work: CJ Amarosa

Credits

Forging new generations of engineers. Moments A lesson on understanding and evaluating moments of...

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