Disc's Angular Velocity Given Circumferential Stress And Outer Radius Calculator

Formula Used:

\[ \omega = \sqrt{\frac{8 \times \sigma_c}{\rho \times \left((3 + \nu) \times r_{outer}^2 - (1 + 3\nu) \times r^2\right)}} \]

Circumferential Stress (σc):

Pascal

Density Of Disc (ρ):

kg/m³

Poisson's Ratio (ν):

Outer Radius Disc (router):

Radius of Element (r):

Angular Velocity (ω):

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1. What Is The Angular Velocity Formula?

The angular velocity formula calculates the rotational speed of a disc based on circumferential stress, material properties, and geometric parameters. It provides insight into the rotational dynamics of solid discs under stress.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ \omega = \sqrt{\frac{8 \times \sigma_c}{\rho \times \left((3 + \nu) \times r_{outer}^2 - (1 + 3\nu) \times r^2\right)}} \]

Where:

\( \omega \) — Angular velocity (rad/s)
\( \sigma_c \) — Circumferential stress (Pa)
\( \rho \) — Density of disc (kg/m³)
\( \nu \) — Poisson's ratio (0.1-0.5)
\( r_{outer} \) — Outer radius of disc (m)
\( r \) — Radius of element (m)

Explanation: The formula relates angular velocity to material stress and properties, accounting for the disc's geometry and material characteristics.

3. Importance Of Angular Velocity Calculation

Details: Calculating angular velocity is crucial for understanding rotational dynamics, designing rotating machinery, and analyzing stress distributions in rotating components.

4. Using The Calculator

Tips: Enter all values in appropriate units. Circumferential stress and density must be positive. Poisson's ratio should be between 0.1-0.5. Outer radius must be greater than element radius.

5. Frequently Asked Questions (FAQ)

Q1: What is angular velocity?
A: Angular velocity measures how fast an object rotates around an axis, expressed in radians per second.

Q2: Why is circumferential stress important?
A: Circumferential stress indicates the tensile stress acting along the circumference of a rotating disc, crucial for structural integrity analysis.

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

Q4: When is this formula applicable?
A: This formula applies to solid rotating discs with uniform material properties and constant angular velocity.

Q5: What are common applications?
A: This calculation is used in mechanical engineering for designing flywheels, turbines, gears, and other rotating machinery components.