Representing a Force and Couple by a Wrench The system in Fig. 4.49 consists of the force and couple F = 3i + 6j + 2k (N), M = 12i + 4j + 6k (N-m). Represent it by a wrench, and determine where the line of action of the wrench's force intersects the x-z plane.

Question

Representing a Force and Couple by a Wrench

The system in Fig. 4.49 consists of the force and couple

F = 3i + 6j + 2k (N),

M = 12i + 4j + 6k (N-m).

Represent it by a wrench, and determine where the line of action of the wrench's force intersects the x-z plane.

Strategy
The wrench is the force F and the component of M parallel to F (Figs, a, b). We must choose the point of application P so that the moment of F about O equals the normal component [latex] M_{n}[/latex]. By letting P be an arbitrary point of the x-z plane, we can determine where the line of action of F intersects that plane.

Accepted Answer

Dividing F by its magnitude, we obtain a unit vector e with the same direction
as F:

[latex] e = \frac{F}{\left|F ight| } = \frac{3i + 6j + 2k}{\sqrt{(3)^{2} +(6)^{2} + (2)^{2} } } =0.429i + 0.857j + 0.286k.[/latex]

We can use e to calculate the component of M parallel to F:

[latex]M_{P} = (e . M)e = [(0.429)(12) + (0.857)(4) + (0.286)(6)]e [/latex]

= 4.408i + 8.816j + 2.939k (N-m).

The component of M normal to F is

[latex]M_{n} = M - M_{P} = 7.592i - 4.816j + 3.061k [/latex] (N-m).

The wrench is shown in Fig. b. Let the coordinates of P be (x, 0, z). The moment of F about O is

[latex]r_{OP} imes F = \left|\begin{matrix}i &j & k \ x & 0 & z \3 &6 & 2 \end{matrix} ight| = -6zi - (2x - 3z)j + 6xk.[/latex]

By equating this moment to [latex]M_{n}[/latex],

-6zi - (2x - 3z)i + 6xk = 7.592i - 4.816j + 3.061k,

we obtain the equations

-6z = 7.592,

-2x + 3z = -4.816,

6x = 3.061.

Solving these equations, we find the coordinates of point P are

X = 0.510 m, z = -1.265 m.

Question 4.21: Representing a Force and Couple by a Wrench The system in Fi...

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