Angular Velocity Of Disc Given Circumferential Stress In Solid Disc Calculator

Angular Velocity Formula:

\[ \omega = \sqrt{\frac{\left(\frac{C_1}{2} - \sigma_c\right) \times 8}{\rho \times r_{disc}^2 \times \left((3 \times \nu) + 1\right)}} \]

Constant At Boundary Condition:

unit

Circumferential Stress:

Pascal

Density Of Disc:

kg/m³

Disc Radius:

meter

Poisson's Ratio:

unit

Angular Velocity:

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1. What Is The Angular Velocity Formula?

The angular velocity formula calculates the rotational speed of a solid disc based on boundary conditions, circumferential stress, material density, disc radius, and Poisson's ratio. This formula is essential in mechanical engineering for analyzing rotating systems.

2. How Does The Calculator Work?

The calculator uses the angular velocity formula:

\[ \omega = \sqrt{\frac{\left(\frac{C_1}{2} - \sigma_c\right) \times 8}{\rho \times r_{disc}^2 \times \left((3 \times \nu) + 1\right)}} \]

Where:

\( \omega \) — Angular velocity (rad/s)
\( C_1 \) — Constant at boundary condition
\( \sigma_c \) — Circumferential stress (Pascal)
\( \rho \) — Density of disc (kg/m³)
\( r_{disc} \) — Disc radius (meter)
\( \nu \) — Poisson's ratio (unitless)

Explanation: The formula relates the rotational characteristics of a solid disc to its material properties and stress conditions.

3. Importance Of Angular Velocity Calculation

Details: Calculating angular velocity is crucial for designing rotating machinery, analyzing stress distributions in rotating components, and ensuring mechanical systems operate within safe rotational speed limits.

4. Using The Calculator

Tips: Enter all required values with appropriate units. Ensure constant, density, and radius are positive values. Poisson's ratio should be between 0 and 0.5 for most materials.

5. Frequently Asked Questions (FAQ)

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

Q2: What is circumferential stress?
A: Circumferential stress is the force per unit area acting tangentially to the circumference of a rotating disc.

Q3: What is Poisson's ratio?
A: Poisson's ratio is the ratio of transverse strain to axial strain when a material is stretched.

Q4: What are typical values for these parameters?
A: Density varies by material (steel: 7850 kg/m³), Poisson's ratio ranges 0.1-0.5, and angular velocity depends on application requirements.

Q5: When is this formula applicable?
A: This formula applies to solid discs with uniform material properties undergoing rotational motion with known boundary conditions.