Equivalent forces
Three cables attached to a disk exert on it the forces shown.
(a) Replace the three forces with an equivalent force-couple system at A.
(b) Determine the single force which is equivalent to the force couple system obtained in part a, and specify its point of application on a line drawn through points A and D.
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Equivalent forces
XB
Y
X
(-100,173.2)(141,141)
(0,-200)
-99i+99j 103.4i+37.6j
99i-99j
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Equivalent forces
XB
Y
X
(-100,173.2)(100,173.2)
(0,-200)
-99i+99j 103.4i+37.6j
99i-99j
FFCC
Equivalent system
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Equivalent forces
FFCC
,
99 99 103.4 37.6 99 99 103.4 37.6(141 141 ) 103.4 37.6
(0 200 ) 99 99 ( 100 173.2 ) 99 995317 14526 19800 9900 1714717838
A
i j i j i j i ji j i j
i j i j i j i jk k k k kk
Z
FM
103.4i+37.6j
17838k
XB
Y
X
(-100,173.2)(141,141)
(0,-200)
-99i+99j
99i-99j
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Equivalent forces
XB
Y
X
(-100,173.2)(141,141)
(0,-200)
-99i+99j 103.4i+37.6j
99i-99j
FF
P (x,y)P (x,y)
Equivalent system
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Equivalent forces
FFP (x,y)P (x,y)
, ( )x y
A x y y x
F i F j
xi yj F i F j xF yF k
Z
F
M
XB
Y
X
(-100,173.2)(141,141)
(0,-200)
-99i+99j
99i-99j
,
99 99 103.4 37.6 99 99 103.4 37.6(141 141 ) 103.4 37.6
(0 200 ) 99 99 ( 100 173.2 ) 99 995317 14526 19800 9900 1714717838
A
i j i j i j i ji j i j
i j i j i j i jk k k k kk
Z
FM
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Equivalent forces
FFP (x,y)P (x,y)
103 4 37 6 103 4 37 6
17838 103 4 37 6 17838
17838
17838
x y x y
y x
F i F j . i . j F . ,F .xF yF . x . y
Equation of line of action Equation of line AD
103.4x-37.6y=x=0
37.6y=- y=-474mm
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Equivalent forces
The steel plate shown will support six 50 mm diameter idler rollers mounted on the plate as shown. Two flat belts pass around the rollers, and rollers A and D will be adjusted so that the tension in each belt is 45 N. Determine
(a) the resultant couple acting on the plate if a = 0.2 m,
(b) The value of a so that the resultant couple acting on the plate is 54 Nm clockwise.
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Equivalent forces
Y
X
(2a+c,0)
(-200,-a)
(-200+c,a+200)
c
c
-45i+0j
45i+0j
-31.82i+31.82j
i aj
a2
45 200
40000
(2a,2a)
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Equivalent forces
Y
X(2a+c,0)
(-200,-a)
(-200+c,a+200)
c
c
-45i+0j
45i+0j
-31.82i+31.82j
i aj. i . j
a 2
45 20031 82 31 82
40000
2
2
45 20045 0 45 0 31.82 31.82
4000045 200
31.82 31.8240000
0
(0 0 ) (31.82 31.82 ) ( 400 0 ) 45 0
( 200 400 ) 45 0 (2 2 ) 31.82 31.820 0 180
c
i aji j i j i j
ai aj
i ja
for a = 200mm
i j i j c i j i j
c i j i j ai aj i jk k
F
M
00 63.86 63.86 18000 255447544
k ak ak k kk
a=200mm
(2a,2a)
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Equivalent forces
Y
X(2a+c,0)
(-200,-a)
(-200+c,2a)
c
c
-45i+0j
45i+0j
-31.82i+31.82j
2
45 200
40000
i aj
a
2
2
2
45 20045 0 45 0 31.82 31.82
4000045 200
31.82 31.8240000
45 200(0 0 ) 2 0 45 0
40000200 2 45 0 (2 2 ) 31.82 31.82
0 0 90 63.86 63.86
c
i aji j i j i j
ai aj
i ja
i aji j a c i j i j
ac i aj i j ai aj i j
k k ak ak ak
F
M
90 127.7290 127.72 5490 73.72
0.82
ak kak k ka
a
(2a,2a)
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