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

Angular Velocity Formula:

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

Circumferential Stress (σc):

Pascal

Density Of Disc (ρ):

kg/m³

Poisson's Ratio (ν):

(unitless)

Outer Radius Disc (router):

meter

Angular Velocity (ω):

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1. What is Angular Velocity of Disc?

Angular velocity of a disc refers to how fast the disc rotates around its central axis. It is measured in radians per second and represents the rate of change of angular displacement with respect to time.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the angular velocity of a solid disc based on its material properties and circumferential stress at the center.

3. Importance of Angular Velocity Calculation

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

4. Using the Calculator

Tips: Enter circumferential stress in Pascals, density in kg/m³, Poisson's ratio (between 0-0.5), and outer radius in meters. All values must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is circumferential stress?
A: Circumferential stress (hoop stress) is the stress exerted circumferentially in a cylindrical or disc-shaped object when subjected to internal or external pressure.

Q2: What is Poisson's ratio?
A: Poisson's ratio is a measure of the Poisson effect, which describes the expansion of material in directions perpendicular to the direction of compression.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.25 and 0.35. Rubber has a Poisson's ratio close to 0.5, while cork is near 0.

Q4: How does disc density affect angular velocity?
A: Higher density generally results in lower angular velocity for the same circumferential stress, as more mass requires more energy to rotate at the same speed.

Q5: What are practical applications of this calculation?
A: This calculation is used in designing flywheels, turbine discs, brake rotors, and other rotating mechanical components where stress and rotational speed are critical factors.