In the given figure,  P=60 lb
  .

The image displays a boom supported by a cable.Cable A C is pin-supported at point A. Boom B C is pin-supported at point B. Point B is directly below point A. Cable A C is inclined at an angle of 65 degrees with the vertical. Boom B C is inclined at an angle of 35 degrees with the vertical. Cable A C is connected to boom B C at point C. 2 forces act at point C. The first force is marked P and it is directed vertically down. The second force has a magnitude of 75 pounds and it is at an angle of 25 degrees with the vertical. The required tension in cable AC is 
100.899
 lb

Determine the corresponding magnitude of the resultant.

The corresponding magnitude of the resultant is 
 lb

We need to determine the magnitude of the resultant force at point C, given the following conditions:

- Force \( P = 60 \, \text{lb} \) is acting vertically downward.
- A second force of \( 75 \, \text{lb} \) is acting at an angle of \( 25^\circ \) with the vertical.
- The tension in cable AC is \( 100.899 \, \text{lb} \).

### Step 1: Break Forces into Components
First, let's break the forces into their horizontal and vertical components.

#### Force P:
Since \( P \) acts vertically downward:
- Horizontal component: \( P_x = 0 \)
- Vertical component: \( P_y = -60 \, \text{lb} \) (negative because it is downward)

#### Second Force (75 lb):
Given the force is at an angle of \( 25^\circ \) with the vertical, we can break it into horizontal and vertical components using trigonometric functions:

- Horizontal component: 
  \[
  F_x = 75 \sin(25^\circ)
  \]
- Vertical component: 
  \[
  F_y = 75 \cos(25^\circ)
  \]

### Step 2: Resultant Components
The total horizontal and vertical components of the resultant force are the sums of the individual components.

- Horizontal component of the resultant:
  \[
  R_x = P_x + F_x = 0 + 75 \sin(25^\circ)
  \]
  
- Vertical component of the resultant:
  \[
  R_y = P_y + F_y = -60 + 75 \cos(25^\circ)
  \]

### Step 3: Magnitude of the Resultant
The magnitude of the resultant force \( R \) can be found using the Pythagorean theorem:

\[
R = \sqrt{R_x^2 + R_y^2}
\]

Let me calculate that for you.The magnitude of the resultant force at point C is approximately \( 32.68 \, \text{lb} \).

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. How would the result change if the angle of the second force was 30 degrees instead of 25 degrees?
2. What effect would doubling the force \( P \) have on the resultant?
3. How would you calculate the tension in the cable if the forces acting were different?
4. What role does the angle between the cable and boom play in this type of analysis?
5. How can you determine the direction of the resultant force?

**Tip**: Always break forces into their horizontal and vertical components to simplify resultant calculations.

In the given figure, P=60 lb. The image displays a boom supported by a cable. Cable A C is pin-supported at point A. Boom B C is pin-supported at point B. Point B is directly below point A. Cable A C is inclined at an angle of 65 degrees with the vertical. Boom B C is inclined at an angle of 35 degrees with the vertical. Cable A C is connected to boom B C at point C. 2 forces act at point C. The first force is marked P and it is directed vertically down. The second force has a magnitude of 75 pounds and it is at an angle of 25 degrees with the vertical. The required tension in cable AC is 100.899 lb. Determine the corresponding magnitude of the resultant.

Solve a physics problem involving a boom and cable system with forces acting at angles. Learn how to break forces into components, use the Pythagorean theorem to determine the resultant force, and apply trigonometric functions. Ideal for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation