Q. 53 - Calculus

Question

A weight of 100 pounds is suspended by two ropes as shown in the accompanying figure. What are the magnitudes of the forces in each of the ropes?

Step-by-Step Solution

Answer

a

Step 1: Introduction:

• Consider a 100-pound weight hanging by two ropes.
• The goal is to determine the force magnitude in each rope.

Step 2: Given Information:

The below given figure shows the weight of 100 pounds suspended by two ropes.

• Because the weight is stationary, the total of the three vectors is zero.
• As a result, $\mathbf{u}+\mathbf{v}+\mathbf{w}=0$ .
• Consider that motion to the right is positive on the $x$-axis, and upward motion is positive on the $y$-axis.
• Decompose each vector into horizontal and vertical components at this point.

Step 3: Explanation

• Horizontal and vertical components are separated.
• $\mathbf{u}=-\cos 45°\Vert \mathbf{u}\Vert \mathbf{i}+\sin 45°\Vert \mathbf{u}\Vert \mathbf{j}$
  
• $\mathbf{v}=\cos 30°\Vert \mathbf{v}\Vert \mathbf{i}+\sin 30°\Vert \mathbf{v}\Vert \mathbf{j}$
  
• $\mathbf{w}=-100\mathbf{j}$
  

Step 4: Explanation

• $\mathbf{u}=-\frac{1}{\sqrt{2}}\Vert \mathbf{u}\Vert \mathbf{i}+\frac{1}{\sqrt{2}}\Vert \mathbf{u}\Vert \mathbf{j}$
  
• $\mathbf{v}=\frac{\sqrt{3}}{2}\Vert \mathbf{v}\Vert \mathbf{i}+\frac{1}{2}\Vert \mathbf{v}\Vert \mathbf{j}\dots \dots \left(3\right)$
  
• $\mathbf{w}=-100\mathbf{j}\dots \dots \left(4\right)$
  

Step 5: Explanation

Using (1), $\mathbf{u}+\mathbf{v}+\mathbf{w}=0$

As a result 

$\mathbf{u}+\mathbf{v}+\mathbf{w}=\left(-\frac{1}{\sqrt{2}}\Vert \mathbf{u}\Vert \mathbf{i}+\frac{1}{\sqrt{2}}\Vert \mathbf{u}\Vert \mathbf{j}\right)+\left(\frac{\sqrt{3}}{2}\Vert \mathbf{v}\Vert \mathbf{i}+\frac{1}{2}\Vert \mathbf{v}\Vert \mathbf{j}\right)-100\mathbf{j}$