Forces $$ F_{1}\left(10 \mathrm{~N}, 090^{\circ}\right), \mathrm{F}_{2}\left(20 \mathrm{~N}, 210^{\circ}\right) $$ and $$ \mathrm{F}_{3}\left(4 \mathrm{~N}, 330^{\circ}\right) $$ act on a particle. Find, correct to one decimal place, the magnitude of the resultant force.

Question

UpStudy AI · Accepted Answer

To find the magnitude of the resultant force acting on the particle, we'll break down each force into its Cartesian components (x and y), sum these components, and then calculate the magnitude of the resulting vector.

### Given Forces:
1. $$ F_{1} = 10 \text{ N at } 90^{\circ} $$
2. $$ F_{2} = 20 \text{ N at } 210^{\circ} $$
3. $$ F_{3} = 4 \text{ N at } 330^{\circ} $$

### Step 1: Break Down Each Force into Components

**Force $$ F_{1} $$:**
- $$ F_{1x} = 10 \cos(90^{\circ}) = 0 $$
- $$ F_{1y} = 10 \sin(90^{\circ}) = 10 \text{ N} $$

**Force $$ F_{2} $$:**
- $$ F_{2x} = 20 \cos(210^{\circ}) = 20 \times (-\frac{\sqrt{3}}{2}) \approx -17.32 \text{ N} $$
- $$ F_{2y} = 20 \sin(210^{\circ}) = 20 \times (-0.5) = -10 \text{ N} $$

**Force $$ F_{3} $$:**
- $$ F_{3x} = 4 \cos(330^{\circ}) = 4 \times \frac{\sqrt{3}}{2} \approx 3.46 \text{ N} $$
- $$ F_{3y} = 4 \sin(330^{\circ}) = 4 \times (-0.5) = -2 \text{ N} $$

### Step 2: Sum of Components
- **Total $$ x $$-component:**
  $$
  F_{\text{total}_x} = F_{1x} + F_{2x} + F_{3x} = 0 - 17.32 + 3.46 \approx -13.86 \text{ N}
  $$
  
- **Total $$ y $$-component:**
  $$
  F_{\text{total}_y} = F_{1y} + F_{2y} + F_{3y} = 10 - 10 - 2 = -2 \text{ N}
  $$

### Step 3: Calculate the Magnitude of the Resultant Force
$$
|\mathbf{F}_{\text{resultant}}| = \sqrt{(F_{\text{total}_x})^2 + (F_{\text{total}_y})^2} = \sqrt{(-13.86)^2 + (-2)^2} \approx \sqrt{191.88 + 4} \approx \sqrt{195.88} \approx 14.0 \text{ N}
$$

### **Final Answer:**
The magnitude of the resultant force is **14.0  N**.
The resultant force has a magnitude of 14.0  N.

Forces and
act on a particle. Find, correct to one decimal place, the
magnitude of the resultant force.

Upstudy AI Solution

Answer

Solution

Given Forces:

Step 1: Break Down Each Force into Components

Step 2: Sum of Components

Step 3: Calculate the Magnitude of the Resultant Force

Final Answer:

Beyond the Answer

Related Questions

Latest Physics Questions

Forces and act on a particle. Find, correct to one decimal place, the magnitude of the resultant force.