Outer Radius Of Disc Given Radial Stress At Center Of Solid Disc Calculator

Formula Used:

\[ r_{outer} = \sqrt{\frac{8 \times \sigma_r}{\rho \times \omega^2 \times (3 + \nu)}} \]

Radial Stress (σr):

Pascal

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

(unitless)

Outer Radius Disc (router):

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1. What is the Outer Radius Calculation?

The outer radius calculation determines the radius of a solid disc based on radial stress, material density, angular velocity, and Poisson's ratio. This is crucial in mechanical engineering for designing rotating discs that can withstand specific stress conditions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{outer} = \sqrt{\frac{8 \times \sigma_r}{\rho \times \omega^2 \times (3 + \nu)}} \]

Where:

\( r_{outer} \) — Outer radius of disc (meters)
\( \sigma_r \) — Radial stress (Pascal)
\( \rho \) — Density of disc (kg/m³)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio (unitless)

Explanation: This formula calculates the maximum outer radius that a solid disc can have while maintaining a specified radial stress at its center, given the material properties and rotational speed.

3. Importance of Outer Radius Calculation

Details: Accurate calculation of outer radius is essential for designing rotating machinery components like flywheels, turbine discs, and gears to ensure they operate within safe stress limits and prevent mechanical failure.

4. Using the Calculator

Tips: Enter radial stress in Pascal, density in kg/m³, angular velocity in rad/s, and Poisson's ratio (typically between 0.1-0.5). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is radial stress in a rotating disc?
A: Radial stress is the stress component acting perpendicular to the radius of the disc, caused by centrifugal forces during rotation.

Q2: Why is Poisson's ratio important in this calculation?
A: Poisson's ratio accounts for the material's tendency to expand or contract in directions perpendicular to the applied stress, affecting the stress distribution.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.25-0.35. Rubber-like materials can have values close to 0.5.

Q4: How does angular velocity affect the outer radius?
A: Higher angular velocities require smaller outer radii to maintain the same radial stress, as centrifugal forces increase with rotational speed.

Q5: Can this formula be used for hollow discs?
A: No, this specific formula is derived for solid discs. Hollow discs have different stress distributions and require different equations.