Disc's Angular Velocity Given Maximum Circumferential Stress In 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):

Outer Radius Disc (router):

meter

Angular Velocity (ω):

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1. What is Angular Velocity in Solid Disc?

Angular velocity in a solid disc refers to the rotational speed at which the disc rotates around its central axis. It's a crucial parameter in mechanical engineering and physics that describes how fast the disc is spinning.

2. How Does the Calculator Work?

The calculator uses the angular velocity 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 (dimensionless)
\( r_{outer} \) — Outer radius of disc (meter)

Explanation: This formula calculates the angular velocity of a solid disc based on the maximum circumferential stress, material properties, and disc geometry.

3. Importance of Angular Velocity Calculation

Details: Calculating angular velocity is essential for designing rotating machinery, analyzing stress distributions 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 (typically between 0.1-0.5), and outer radius in meters. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is circumferential stress in a rotating disc?
A: Circumferential stress (hoop stress) is the stress acting along the circumference of the disc caused by rotational forces.

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.

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 has around 0.5, while cork is close to 0.

Q4: How does disc density affect angular velocity?
A: Higher density materials require lower angular velocities to achieve the same circumferential stress, as density appears in the denominator of the formula.

Q5: What are the limitations of this formula?
A: This formula assumes homogeneous material properties, constant thickness, and elastic behavior within the material's limits.