The three similar torque ( `tau`) are acting at an angle of 120`degree` with each other. The resultant torque will be?

Options

A. zero
B. `sqrt2tau`
C. `3tau`
D. `tau/3`

Answer

To find the resultant torque when three similar torques (τ) are acting at an angle of 120 degrees with each other, we can follow these steps:

### Step 1: Understand the Configuration
We have three torque vectors of equal magnitude (τ) acting at angles of 120 degrees relative to each other.

### Step 2: Represent the Torques as Vectors
We can represent these torques as vectors in a 2D coordinate system. Let's denote the first torque vector as τ₁, the second as τ₂, and the third as τ₃.

- Torque τ₁ can be represented along the positive x-axis.
- Torque τ₂ will be at an angle of 120 degrees from τ₁.
- Torque τ₃ will be at an angle of 240 degrees from τ₁ (or 120 degrees from τ₂).

### Step 3: Calculate the Components of Each Torque
Using trigonometric functions, we can break each torque vector into its horizontal (x) and vertical (y) components.

1. **Torque τ₁:**
- τ₁ (x-component) = τ
- τ₁ (y-component) = 0

2. **Torque τ₂ (120 degrees):**
- τ₂ (x-component) = τ * cos(120°) = τ * (-1/2) = -τ/2
- τ₂ (y-component) = τ * sin(120°) = τ * (√3/2) = τ√3/2

3. **Torque τ₃ (240 degrees):**
- τ₃ (x-component) = τ * cos(240°) = τ * (-1/2) = -τ/2
- τ₃ (y-component) = τ * sin(240°) = τ * (-√3/2) = -τ√3/2

### Step 4: Sum the Components
Now, we can sum the components in both the x and y directions.

**Horizontal (x-direction):**
\[ R_x = τ + (-τ/2) + (-τ/2) = τ - τ = 0 \]

**Vertical (y-direction):**
\[ R_y = 0 + (τ√3/2) + (-τ√3/2) = τ√3/2 - τ√3/2 = 0 \]

### Step 5: Calculate the Resultant Torque
The resultant torque (R) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} \]
Since both \( R_x \) and \( R_y \) are zero:
\[ R = \sqrt{0^2 + 0^2} = 0 \]

### Conclusion
The resultant torque when three similar torques are acting at an angle of 120 degrees with each other is **0**.