Answer :
Answer:
Explanation:F as the magnitude of the couple at A.
�
1
r
1
as the distance from point A to point B.
�
2
r
2
as the distance from point A to point C.
The moment produced by the wrench at B can be calculated as the force
�
�
F
B
applied by the wrench multiplied by the perpendicular distance from point B to the line of action of the force, denoted as
�
1
r
1
.
The moment produced by the couple at A is equal to the magnitude of the couple
�
F multiplied by the distance
�
2
r
2
.
For simplification, these moments should be equal and opposite.
So, we have:
�
�
⋅
�
1
=
�
⋅
�
2
F
B
⋅r
1
=F⋅r
2
We need to solve for
�
F.
But first, we need to find
�
�
F
B
. To do that, we need to resolve the force applied by the wrench into its horizontal and vertical components.
Let's assume:
The force applied by the wrench is
�
�
F
W
.
�
θ is the angle between the wrench and the horizontal.
Then, the horizontal component
�
�
�
F
Bx
of the force
�
�
F
B
is given by
�
�
⋅
cos
(
�
)
F
W
⋅cos(θ), and the vertical component
�
�
�
F
By
is given by
�
�
⋅
sin
(
�
)
F
W
⋅sin(θ).
Since the system is in equilibrium, the vertical component
�
�
�
F
By
must cancel out the vertical component of the force at point A.
Given that the force at point A is perpendicular to the lever arm,
�
�
�
F
By
can be equated to
�
F, the magnitude of the couple force.
So,
�
�
�
=
�
F
By
=F.
Now, we can find the horizontal component
�
�
�
F
Bx
using the equilibrium condition:
�
�
�
=
�
�
⋅
cos
(
�
)
F
Bx
=F
W
⋅cos(θ)
Now, we can calculate the moment
�
�
M
B
produced by the wrench at point B:
�
�
=
�
�
�
⋅
�
1
M
B
=F
Bx
⋅r
1
Finally, we set
�
�
M
B
equal to the moment produced by the couple at point A:
�
�
=
�
⋅
�
2
M
B
=F⋅r
2
Solve this equation for
�
F, and you'll find the magnitude of the couple force needed.
