Lecture # 3.pdf, Moment of Force and Resultant

Resultant of a moment

1. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
Beams are often used to bridge gaps in walls.
We have to know what the effect of the force
on the beam will have on the supports of the
beam.
What do you think is happening at points A and B?
MOMENT OF A FORCE

2. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
Carpenters often use a hammer in this way to pull a
stubborn nail. Through what sort of action does the force
F
H at the handle pull the nail? How can you mathematically
model the effect of force F
H at point O?

3. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
MOMENT OF A FORCE
When a force is applied on a body it will produce a tendency
for the body to rotate about a point that is not on the line of
action of the force.
This tendency to rotate is sometimes called a torque. but most
often it is called the moment of a force or simply the Moment.
The magnitude of the moment is
directly proportional to the
magnitude of F and the perpendicular
distance or moment arm d. The
larger the force or the longer the
moment arm, the greater the moment or
turning effect.

4. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
MOMENT OF A FORCE (continued)
As shown, d is the moment arm or perpendicular distance from point O to the
line of action of the force.
In 2-D, the direction of M
O is either clockwise (CW) or counter-clockwise
(CCW), depending on the tendency for rotation.
In a 2-D case, the magnitude of the moment is M
o = F d
Units of moment magnitude
consist of force times distance.
e.g. N. m or lb. fl.

5. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
MOMENT OF A FORCE (continued)
Often it is easier to determine M
O by
using the components of F as shown.
Then M
O = (F
Y * a) – (F
X * b).
The typical sign convention for a moment in 2-D is that counter-
clockwise is considered positive.
For example, M
O = F * d
and the direction is counter-clockwise.
F
a
b
d
O
a
b
O
F
F
x
F
y

6. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
Resultant Moment

For two-dimensional problems. where all the forces lie within the
X – Y plane, In Fig. below, the resultant moment (M
R), about point
o (the z- axis) can be determined by finding the algebraic sum of the
moments caused by all the forces in the system. As a convention we
will generally consider positive moments as counter clockwise since
they are directed along the positive z axis (out of the page).

7. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
Resultant Moment.
Clockwise moments will be negative. Doing this, the directional
sense of each moment can be represented by a plus or minus sign.
Using this sign convention, the resultant moment in Fig. 4--3 is
therefore



If the numerical result of this sum is a positive, (M
R) will
be a counter clockwise moment (out of the page): and if the
result is negative, (M
R) will be a clockwise moment (into
the page).

8. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.

9. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.

10. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.

11. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
The moment of a force does not always cause a rotation.
For example: the force F tends to rotate the beam clockwise
about its support at A with a moment M
A

= F . d
A.
The actual rotation would occur if the support at B were
removed.

12. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
Principle of Moments

A concept often used in mechanics is the principle of moment,
which is sometimes referred to as Varignon’s theorem since it was
originally developed by the French mathematician Varignon
(1654- 1722).
It states that moment of a force about a point is equal to the sum of
the moments of the components of the force about the point.

13. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
For two dimensional problems, we can use the principle of
moments by resolving the force into its rectangular components
and then determine the moment using a scalar analysis.

14. Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.