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 are acting at an angle of 120 degrees with each other, we can follow these steps:

### Step 1: Understand the Torque Configuration
We have three torques (τ) acting at angles of 120 degrees relative to each other. We can visualize this by placing one torque along the horizontal axis.

### Step 2: Draw the Torque Vectors
1. Let the first torque (τ₁) be along the positive x-axis.
2. The second torque (τ₂) will be at an angle of 120 degrees from τ₁.
3. The third torque (τ₃) will be at an angle of 240 degrees from τ₁ (or 120 degrees from τ₂).

### Step 3: Break Down the Torques into Components
For each torque, we can break them down into their horizontal (x-axis) and vertical (y-axis) components.

- **Torque τ₁:**
- τ₁ = τ (acting along the x-axis)
- Horizontal component: τ₁x = τ
- Vertical component: τ₁y = 0

- **Torque τ₂ (120 degrees from τ₁):**
- Horizontal component: τ₂x = τ * cos(120°) = τ * (-1/2) = -τ/2
- Vertical component: τ₂y = τ * sin(120°) = τ * (√3/2)

- **Torque τ₃ (240 degrees from τ₁):**
- Horizontal component: τ₃x = τ * cos(240°) = τ * (-1/2) = -τ/2
- Vertical component: τ₃y = τ * sin(240°) = τ * (-√3/2)

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

- **Total Horizontal Component (Στx):**
\[ Στx = τ + (-τ/2) + (-τ/2) = τ - τ = 0 \]

- **Total Vertical Component (Στy):**
\[ Στy = 0 + (τ * √3/2) + (-τ * √3/2) = 0 \]

### Step 5: Calculate the Resultant Torque
The resultant torque (τ_net) can be calculated using the Pythagorean theorem:
\[ τ_{net} = \sqrt{(Στx)^2 + (Στy)^2} = \sqrt{0^2 + 0^2} = 0 \]

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

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