Angular Velocity Of Disc Given Radial Stress At Center Of Solid Disc Calculator

Formula Used:

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

Radial Stress (\(\sigma_r\)):

Density Of Disc (\(\rho\)):

kg/m³

Poisson's Ratio (\(\nu\)):

Outer Radius Disc (\(r_{outer}\)):

Angular Velocity (\(\omega\)):

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1. What is Angular Velocity Of Disc Given Radial Stress?

The angular velocity of a disc given radial stress at the center calculates the rotational speed of a solid disc based on the radial stress, material properties, and disc dimensions. This is important in mechanical engineering applications involving rotating discs.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the angular velocity at which a solid disc rotates based on the radial stress at its center, considering the material properties and disc geometry.

3. Importance of Angular Velocity Calculation

Details: Calculating angular velocity from radial stress is crucial for designing rotating machinery, analyzing stress distributions in rotating components, and ensuring structural integrity under rotational forces.

4. Using the Calculator

Tips: Enter radial stress in Pascals, density in kg/m³, Poisson's ratio (typically between 0.1-0.5), and outer radius in meters. All values must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is radial stress in a rotating disc?
A: Radial stress is the stress acting along the radius of the disc, perpendicular to the circumferential direction, 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 in the disc.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.1 and 0.5, with common values around 0.3 for many engineering materials.

Q4: Can this formula be used for hollow discs?
A: This specific formula is derived for solid discs. Different equations apply for hollow discs with different boundary conditions.

Q5: What units should be used for input values?
A: Use Pascals for stress, kg/m³ for density, dimensionless for Poisson's ratio, and meters for radius to ensure consistent SI units.