Upload
etcenterrbru
View
592
Download
3
Tags:
Embed Size (px)
Citation preview
Moment of a force about a point or axis – a
measure of the tendency of the force to
cause a body to rotate about the point or axis
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
cause a body to rotate about the point or axis
Case 1
Consider horizontal force Fx,
which acts perpendicular to
the handle of the wrench and
is located dy from the point O
� Fx tends to turn the pipe about the z axis
� The larger the force or the distance dy, the greater the turning effect
� Torque – tendency of
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
� Torque – tendency of
rotation caused by Fx
or simple moment (Mo) z
� Moment axis (z) is perpendicular to shaded
plane (x-y)
� Fx and dy lies on the shaded plane (x-y)
Moment axis (z) intersects
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
� Moment axis (z) intersects
the plane at point O
Case 2
Apply force Fz to the wrench
� Pipe does not rotate about z axis
Tendency to rotate about x axis
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
� Tendency to rotate about x axis
� The pipe may not actually
rotate Fz creates tendency
for rotation so moment
(Mo) x is produced
Case 2
� Moment axis (x) is perpendicular to shaded
plane (y-z)
F and d lies on the shaded plane (y-z)
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
� Fz and dy lies on the shaded plane (y-z)
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Case 3
Apply force Fy to the wrench
� No moment is produced about point O
Lack of tendency to rotate � Lack of tendency to rotate
as line of action passes
through O
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
In General
� Consider the force F and the point O which lies in the shaded plane
� The moment M about point O, � The moment MO about point O,
or about an axis passing
through O and perpendicular
to the plane, is a vector quantity
� Moment MO has its specified
magnitude and direction
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Magnitude� For magnitude of MO,
MO = FdO
where d = moment arm or perpendicular distance from the axis at point O to its line of action of the force
� Units for moment is N.m
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Direction
� Direction of MO is specified by using “right hand rule”using “right hand rule”
- fingers of the right hand are curled to follow the sense of rotation when force rotates about point O
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Direction
- Thumb points along the moment axis to give the moment axis to give the direction and sense of the moment vector
- Moment vector is upwards and perpendicular to the shaded plane
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Direction
MO is shown by a vector arrow
with a curl to distinguish it from
force vector
Example (Fig b)
� MO is represented by the counterclockwise curl, which indicates the action of F
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Direction
� Arrowhead shows the sense of rotation caused by F
Using the right hand rule, the � Using the right hand rule, the direction and sense of the moment vector points out of the page
� In 2D problems, moment of the force is found about a point O
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Direction
� Moment acts about an axis perpendicular to the plane containing F and dcontaining F and d
� Moment axis intersects
the plane at point O
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Resultant Moment of a System of
Coplanar Forces
� Resultant moment, MRo = addition of the � Resultant moment, MRo = addition of the moments of all the forces algebraically since all moment forces are collinear
MRo = ∑Fd
taking clockwise to be positive
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Resultant Moment of a System of
Coplanar Forces
� A clockwise curl is written along the equation � A clockwise curl is written along the equation to indicate that a positive moment if directed along the
+ z axis and negative
along the – z axis
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
� Moment of a force does not always cause rotation
� Force F tends to rotate the beam clockwise about A with moment
MA = FdAMA = FdA
� Force F tends to rotate the beam counterclockwise about B with moment
MB = FdB
� Hence support at A prevents
the rotation
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Example 4.1
For each case, determine the moment of the
force about point Oforce about point O
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Solution
� Line of action is extended as a dashed line to establish moment arm d
� Tendency to rotate is indicated and the orbit is � Tendency to rotate is indicated and the orbit is shown as a colored curl
)(.5.37)75.0)(50()(
)(.200)2)(100()(
CWmNmNMb
CWmNmNMa
o
o
==
==
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Solution
)(.0.21)14)(7()(
)(.4.42)45sin1)(60()(
)(.229)30cos24)(40()(
CCWmkNmmkNMe
CCWmNmNMd
CWmNmmNMc
o
o
=−=
==
=+=
o
o
)(.0.21)14)(7()( CCWmkNmmkNMe o =−=
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Example 4.2
Determine the moments of
the 800N force acting on the
frame about points A, B, C frame about points A, B, C
and D.
4.1 Moment of a Force –Scalar Formation
4.1 Moment of a Force –Scalar Formation
Solution
Scalar Analysis
)(.1200)5.1)(800(
)(.2000)5.2)(800(
CWmNmNM
CWmNmNM A
==
==
Line of action of F passes through C
)(.400)5.0)(800(
.0)0)(800(
)(.1200)5.1)(800(
CCWmNmNM
mkNmNM
CWmNmNM
D
C
B
==
==
==
4.1 Moment of a Force – Scalar Formation
4.1 Moment of a Force – Scalar Formation
Example 4.3
Determine the resultant moment of the four
forces acting on the rod about point Oforces acting on the rod about point O
4.1 Moment of a Force –Scalar Formation
4.1 Moment of a Force –Scalar Formation
Solution
Assume positive moments acts in the +k
direction, CCW
FdM ∑=
)(.334
.334
)30cos34)(40(
)30sin3)(20()0)(60()2)(50(
CWmN
mN
mmN
mNmNmNM
FdM
Ro
Ro
=
−=
+−
++−=
∑=
o
o