The weight of a crate is 300 newtons. Find the tension in each of the supporting cables shown in the figure. The coordinates of the points $A , B , C$, and $D$ are given below the figure. Round to the nearest newton.

[Figure not necessarily to scale.]
point $A = ( 0,0 , - 130 )$, point $B = ( 90,0,0 )$, point $C = ( - 40,40,0 )$, point $D = ( 0 , - 180,0 )$
A) cable $A B = 196$; cable $A C = 97$; cable $A D = 68$
B) cable $A B = 97$; cable $A C = 68$; cable $A D = 196$
C) cable $A B = 97$; cable $A C = 196$; cable $A D = 68$
D) cable $A B = 68$; cable $A C = 196$; cable $A D = 97$
E) cable $A B = 196$; cable $A C = 68$; cable $A D = 97$

Question

The weight of a crate is 300 newtons. Find the tension in each of the supporting cables shown in the figure. The coordinates of the points $A , B , C$, and $D$ are given below the figure. Round to the nearest newton.

[Figure not necessarily to scale.]
point $A = ( 0,0 , - 130 )$, point $B = ( 90,0,0 )$, point $C = ( - 40,40,0 )$, point $D = ( 0 , - 180,0 )$
A) cable $A B = 196$; cable $A C = 97$; cable $A D = 68$ 
B) cable $A B = 97$; cable $A C = 68$; cable $A D = 196$ 
C) cable $A B = 97$; cable $A C = 196$; cable $A D = 68$ 
D) cable $A B = 68$; cable $A C = 196$; cable $A D = 97$ 
E) cable $A B = 196$; cable $A C = 68$; cable $A D = 97$

Quizplus · Accepted Answer

We can use the method of resolving forces into components. For this, we first find the unit vectors along each cable by dividing the vector with the magnitude of the cable. $\vec{AB} = <90, 0, 130>$ and $|\vec{AB}| = \sqrt{90^2+130^2}=196$ Unit vector along $\vec{AB}$ is $\hat{u}_{AB} = <\frac{9}{14}, 0, \frac{13}{14}>$ Tension in cable AB = $300 imes \hat{u}_{AB, z} = 300 imes \frac{13}{14} \approx 276 N$ $\vec{AC} = <-40, 40, 130>$ and $|\vec{AC}| = \sqrt{(-40)^2+40^2+130^2}=196$ Unit vector along $\vec{AC}$ is $\hat{u}_{AC} = <-\frac{2\sqrt{2}}{7}, \frac{2\sqrt{2}}{7}, \frac{3\sqrt{2}}{7}>$ Tension in cable AC = $300 imes \hat{u}_{AC, z} = 300 imes \frac{3\sqrt{2}}{7} \approx 195 N$ $\vec{AD} = <0, -180, 130>$ and $|\vec{AD}| = \sqrt{180^2+130^2}=221$ Unit vector along $\vec{AD}$ is $\hat{u}_{AD} = <0, -\frac{180}{221}, \frac{130}{221}>$ Tension in cable AD = $300 imes \hat{u}_{AD, z} = 300 imes \frac{130}{221} \approx 177 N$ Rounding each answer to the nearest Newton, we get cable AB as 276 N, cable AC as 195 N, and cable AD as 177 N. Therefore, the answer is C.

The Weight of a Crate Is 300 Newtons A,B,CA , B , CA,B,C

The Weight of a Crate Is 300 Newtons $A , B , C$