Determining Components of a Force The cable AB in Fig. 2.31 exerts a 50-N force T on the collar at A. Express T in terms of components.

Question

Determining Components of a Force

The cable AB in Fig. 2.31 exerts a 50-N force T on the collar at A. Express T in terms of components.

Strategy
Let [latex]r_{AB} [/latex] be the position vector from A to B. We will divide [latex]r_{AB} [/latex] by its magnitude to obtain a unit vector [latex]e_{AB} [/latex] having the same direction as the force T. Then we can obtain T in terms of scalar components by expressing it as the product of its magnitude and [latex]e_{AB} [/latex]. To begin this procedure, we must first determine the coordinates of the collar A. We will do so by obtaining a unit vector [latex]e_{CD} [/latex] pointing from C toward D and multiplying it by 0.2 m to determine the position of the collar A relative to C.

Accepted Answer

Determining the Coordinates of Point A The position vector from C to D is

[latex]r_{CD} = (0.2 - 0.4)i + (0 - 0.3)j + (0.25 - 0)k [/latex]

= - 0.2i - 0.3j + 0.25k (m).

Dividing this vector by its magnitude, we obtain the unit vector [latex]e_{CD} [/latex] (Fig- a):

[latex]e_{CD} = \frac{r_{CD} }{\left|r_{CD} \right|} = \frac{- 0.2i - 0.3j + 0.25k}{\sqrt{(- 0.2)^{2} +(- 0.3)^{2} + (0.25)^{2} } }[/latex]

= - 0.456i - 0.684j + 0.570k.

Using this vector, we obtain the position vector from C to A:

[latex]r_{CA} = (0.2 m)e_{CD} = - 0.091i - 0.137j + 0.114k (m). [/latex]

The position vector from the origin of the coordinate system to C is [latex]r_{OC} = 0.4 i + 0.3j (m)[/latex] , so the position vector from the origin to A is

[latex]r_{OA} = r_{OC} + r_{CA} = (0.4i + 0.3j) + (-0.091 i - 0.137j + 0.114k) = 0.309i + 0.163j + 0.114k (m).[/latex]

The coordinates of A are (0.309, 0.163, 0.114) m.

Determining the Components of T Using the coordinates of point A, the position vector from A to B is

[latex]r_{AB} =(0 - 0.309)i + (0.5 - 0.163)j + (0.15 - 0.114)k = -0.309i + 0.337j + 0.036k (m) [/latex] .

Dividing this vector by its magnitude, we obtain the unit vector [latex]e_{AB} [/latex] [Fig. (a)]:

[latex] e_{AB} = \frac{r_{AB} }{\left|r_{AB}\right| } = \frac{-0.309i + 0.337j + 0.036k}{\sqrt{(- 0.309)^{2} + (0.337)^{2} + (0.036)^{2} } } = -0.674i + 0.735j + 0.079k.[/latex]

The force T is

[latex]T = \left|T\right| e_{AB} = (50 N)(-0.674i + 0.735j + 0.079k) = -33.7i + 36.7j + 3.9k (N) [/latex].

Question 2.12: Determining Components of a Force The cable AB in Fig. 2.31 ...

Related Answered Questions

Determining Scalar Components The tether of the balloon in Fig. 2.28 exerts an 800-N force F on the hook at O. The vertical line AB intersects the x-z plane at point A. The angle between the z axis and the line OA is 60°, and the angle between the line OA and F is 45°. Express F in terms of ...

Verified Answer:

Determining Scalar Components The crane in Fig. 2.27 exerts a 600-lb force F on the caisson. The angle between F and the x axis is 54°, and the angle between F and the y axis is 40°. The z component of F is positive. Express F in terms of components. Strategy Only two of the angles between the ...

Verified Answer:

Verified Answer:

Verified Answer:

Vector Whose Direction is Specified by Two Points The bar AB in Fig. 2.29 exerts a 140-N force F on its support at A. The force is parallel to the bar and points toward B. Express F in terms of components. ...

Verified Answer:

Components Parallel and Normal to a Line Suppose that you pull on the cable OA in Fig. 2.37, exerting a 50-N force F at O. What are the components of F parallel and normal to the cable OB? ...

Verified Answer:

Verified Answer:

Verified Answer:

Determining Components in Three Dimensions The rope in Fig. 2.30 extends from point B through a metal loop attached to the wall at A to point C. The rope exerts forces FAB and FAC on the loop at A with magnitudes | FAB | = | FAB | = 200 lb. What is the magnitude of the total force F = FAB +FAC ...

Verified Answer:

Using the Dot Product to Determine an Angle What is the angle θ between the lines AB and AC in Fig. 2.36? ...

Verified Answer: