https://r2.erweima.ai/mathbot/image/1724722270469repn2c69.jpg

Two forces F1 = 100 N and F2 = 150 N, act on the hook shown according to the angles indicated. Determine the magnitude and direction of the resultant force on the hook.


To determine the magnitude and direction of the resultant force acting on the hook, we will break down each force into its components along the x-axis and y-axis and then sum these components to find the resultant force.

### Step 1: Resolve Forces into Components
Given:
- \( F_1 = 100 \, \text{N} \) at an angle of \( 15^\circ \) below the x-axis
- \( F_2 = 150 \, \text{N} \) at an angle of \( 10^\circ \) above the y-axis

#### Components of \( F_1 \):
- \( F_{1x} = F_1 \cos(15^\circ) \)
- \( F_{1y} = -F_1 \sin(15^\circ) \) (negative because it's below the x-axis)

#### Components of \( F_2 \):
- \( F_{2x} = F_2 \sin(10^\circ) \) (since the angle is with the y-axis)
- \( F_{2y} = F_2 \cos(10^\circ) \)

### Step 2: Calculate Components
Let's calculate the numerical values:

- \( F_{1x} = 100 \times \cos(15^\circ) \)
- \( F_{1y} = -100 \times \sin(15^\circ) \)
- \( F_{2x} = 150 \times \sin(10^\circ) \)
- \( F_{2y} = 150 \times \cos(10^\circ) \)

We can calculate these values now.

### Step 3: Find the Resultant Components
- \( R_x = F_{1x} + F_{2x} \)
- \( R_y = F_{1y} + F_{2y} \)

### Step 4: Magnitude and Direction of the Resultant Force
- Magnitude: \( R = \sqrt{R_x^2 + R_y^2} \)
- Direction: \( \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) \)

I'll calculate these steps now.The magnitude of the resultant force acting on the hook is approximately **172.87 N**, and its direction is approximately **44.81°** above the positive x-axis.

Do you want more details or have any questions?

Here are some related questions you might find helpful:
1. How do you resolve forces into their x and y components?
2. Why is the direction of the resultant force calculated using the arctan function?
3. What would happen if the angles of the forces were different?
4. How would the problem change if a third force was introduced?
5. What is the significance of the direction of the resultant force?

**Tip:** Always double-check the sign of the components when resolving forces, especially when angles are given with respect to different axes.

Learn how to determine the magnitude and direction of the resultant force acting on a hook when two forces F1 = 100 N and F2 = 150 N are applied at specific angles. This problem involves resolving forces into components, calculating magnitudes, and determining directions using trigonometric principles. Suitable for advanced high school students and those studying physics or engineering.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation