4.3 Moment of a Couple - School of · PDF file4.3 Moment of a Couple Example 1, page 1 of 3 1. Two swimmers on opposite sides of a boat attempt to turn the boat by pushing as shown

4.3 Moment of a Couple

4.3 Moment of a Couple Example 1, page 1 of 3
1. Two swimmers on opposite sides of a boat
attempt to turn the boat by pushing as shown.
Determine the couple moment about
a) point A on the bow,
b) point B on the stern, and
c) point C.
Also state what general principles your results
demonstrate.
50 N
B
C
A
50 N
1 m
1.5 m
2 m

ns.
ns.
21 Calculate the moment about A.
MA = (50 N)(1 m) (50 N)(1 m + 2 m)
= 100 Nm
Calculate the moment about B.
MB = (50 N)(2 m + 1.5 m) (50 N)(1.5 m)
= 100 Nm
1.5 m
50 NC
B
50 N50 N
C
A
50 N
1 m
2 m 2 m

ns.
3 Moment about C
MC = (50 N)(2 m)
= 100 Nm
ns.
General principles demonstrated:
1) The couple moment is the same about every
point, and
2) The calculation of the couple moment is
simplified if the moment is calculated about a
point on the line of action of one of the forces
making up the couple.
4
50 N
2 m
C
50 N

30 N
Resolve the inclined forces into rectangular components.
1
20 N
30 N
20 N
A D
CB
B C
DA
20 N 30 N
20 N30 N
(20 N) sin30= 10 N
(20 N) cos 30= 17.32 N
30
(20 N) sin 30= 10 N
(20 N) cos 30= 17.32 N
30
4 m
8 m30
2. Determine the magnitude and sense of the resultant couple
moment acting on the rectangular plate.
30

30 N
8 m
10 N
30 N10 N
A D
CB
Moment of the couple formed by the forces at A and C.
MC = (17.32 N)(4 m) (10 N)(8 m)
= 10.72 Nm (2)
This moment value would be the same for any other point besides C.
Moment of the couple formed by the forces at B and D.
MD = (30 N)(8 m)
= 240 Nm (1)
This moment value would be the same for any other point
besides D.
3
4
2 Calculate the moments
17.32 N
17.32 N
4 m

A D
CB
Arbitrary location
Since both couple moments are the same about all
points, we can move them to any arbitrary point
we choose and then add them to get the resultant
couple moment.
M = MC + MD
= 10.72 + 240
= 229 Nm
5
6
No subscript because
valid for all points
ns.
229 Nm

3. Determine the value of the force P such that the
resultant couple moment of the two couples acting on
the beam is 900 lbft clockwise.
200 lb
200 lbPP
E
C
DB
A
5 ft
3
45
345 0.5 ft
7 ft4 ft

5 ft 7 ft
A
B D
C
E
160 lbP P
ns.
2 Compute the moment of the couple at C and E.
ME = (120 lb)(0.5 ft) + (160 lb)(5 ft + 7 ft)
= 1860 lbft (2)
3
Since the resultant couple moment is specified to
be 900 lbft clockwise, Eq. 3 becomes
900 lbft = P(5) + 1860
Solving gives
P = 552 lb
Since the couple moments are the same about all
points, we can move them both to any arbitrary
point we choose and then add them to get the
resultant couple moment:
M = P(5) + 1860 (3)
4
1 Compute the moment of the couple at B and D.
MD = P(5 ft) (1)
120 lb
160 lb
120 lb
0.5 ft

C
10 Nm
4. The wrench applies a 10 Nm couple
moment to the bolt. To prevent the plate
from rotating, two 2-N forces are applied as
shown. Determine the distance s such that the
resultant couple moment acting on the plate
and bolt is zero.
2 N 2 N
BA
s

To calculate the moment of the couple, sum the moments of
the two forces at A and B with respect to the point A:
MA = (2 N)s
Since couple moments are the same about any point, the two
couple moments of (2 N)s and 10 Nm can be considered to act
at the same point and thus added to give the resultant moment,
M (Note that no subscript is needed on M, since M is
independent of where the moment is calculated):
M = (2 N)s 10 Nm
1
2
Because the resultant moment, M, is to be zero, the latter
equation becomes
0 = (2 N)s 10
Solving gives
s = 5 m
3
ns.
s
A B
2 N2 N
10 Nm
C

P
5. Two cords are wrapped around pegs attached to a
board as shown. Determine the value of such that P is
as small as possible while still producing a resultant
couple moment of zero. Also determine the value of P
corresponding to this value of . Neglect the size of the pegs.
2 m
A B
DC
80 N
P
80 N
4 m

Moment of the couple at B and C.
A
D
1
MC = (80 N)(2 m)
= 160 Nm (1)
2 To balance the 160 Nm couple moment computed in
Eq. 1, the moment of the couple at A and D must be
160 Nm. To achieve this value of couple moment
with the smallest possible value of force P, P must be
perpendicular to line AD
B
80 N
80 N
PP
C
2 m

Now use geometry to find .
A
D
3
4 Equal angles
PP
tan-1
= 63.4 (2)
52 m4 m
C
4 m
distance AD =
= 4.472 m
6
7 Moment of couple A and D
MD = P distance AD
= P(4.472 m) (3)
(2 m)2 + (4 m)2
ns.
C
P P
D
A
B
2 m
2 m
4 m

8 The resultant couple moment must be zero:
M = MC + MD = 0 (4)
or,
160 + P(4.472 m) = 0
Solving gives
P = 35.8 N
Alternative solution. Do not assume that force P is
perpendicular to line AD. Calculate the couple moment of P
about point D in terms of the unknown angle .
ns.
9
P cos
P sin
A
D
PP
80cos + 2 sin
MD = (P cos )(2 m) + (P sin )(4 m) (5)
Now substitute the expressions for MC and MD into Eq. 4:
MC + MD = 0 (Eq. 4 repeated)
by Eq. 1 by Eq. 5
160 (P cos )(2) + (P sin )(4)
Solving for P gives
P = (6)
4 m
2 m

dPd
= 0
From Eq. 6,
= ddP d
d 80cos + 2 sin
= 0
= 0(cos + 2 sin )2
80 ( sin + 2 cos )
Thus
sin + 2 cos = 0
tan = 2
= tan-1 2
= 63.4
This is the same result as Eq. 2.
so
10 To find the minimum value of P, use

250 mm130 mm
175 mm
6. A plumber uses two pipe wrenches so that he can
loosen pipe BC from pipe AB without also
loosening pipe AB from the connection at the wall,
A. Determine the moment of the forces about a) A
and b) D. Also state what general principles your
results demonstrate.
A
B C
D
z
x
y
80 N
80 N
E

z175 mm
250 mm130 mm
rBDrAD
E80 N
80 N
A
y
Equal magnitude, opposite
sign, so cancel out
ij
k
rAD = {130i 175j} mm (1)
rAE = {130 mm + 250 mm}i (175 mm}j
= {380i 175j} mm (2)
MA = rAD {80k} N + rAE { 80k} N
= {130i 175j} {80k} + {380i 175j} { 80k}
= 130(80)i k 175(80)j k + 380( 80)i k 175( 80)j k
= j = j
= [130(80) + 380( 80)]( j)
= {20 000j} Nmm
= {20j} Nm
2 Use the cross product definition of moment.
Part a): Determine the moment about
point A. Introduce position vectors
from point A to points D and E
respectively.
1
x
D
CB

z 175 mm
250 mm
E
80 N
80 N
D
A
y
6
rDE
MD
The two forces (the couple) produce a moment that tends to rotate the
entire pipe assembly about the vertical axis (j component). The force
applied at E also produces a moment on pipe BC about the x axis, while
the

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4.3 Moment of a Couple - School of · PDF file4.3 Moment of a Couple Example 1, page 1 of 3 1. Two swimmers on opposite sides of a boat attempt to turn the boat by pushing as shown