Outer Radius Of Disc Given Circumferential Stress At Center Of Solid Disc Calculator

Formula Used:

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

Circumferential Stress (σc):

Pascal

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

(unitless)

Outer Radius Disc (router):

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1. What is the Outer Radius Calculation Formula?

The formula calculates the outer radius of a solid disc based on circumferential stress, material density, angular velocity, and Poisson's ratio. This calculation is essential in mechanical engineering for designing rotating discs and analyzing stress distribution.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

Where:

\( r_{outer} \) — Outer radius of the disc (meters)
\( \sigma_c \) — Circumferential stress (Pascal)
\( \rho \) — Density of the disc material (kg/m³)
\( \omega \) — Angular velocity (radians per second)
\( \nu \) — Poisson's ratio (unitless)

Explanation: The formula derives from the equilibrium equations for rotating discs and accounts for the material properties and rotational dynamics.

3. Importance of Outer Radius Calculation

Details: Accurate calculation of outer radius is crucial for designing rotating mechanical components such as flywheels, turbine discs, and gears. It ensures structural integrity under rotational stresses and prevents failure due to excessive deformation or stress concentrations.

4. Using the Calculator

Tips: Enter circumferential stress in Pascals, density in kg/m³, angular velocity in rad/s, and Poisson's ratio (typically between 0.1-0.5 for metals). All values must be positive and within valid ranges.

5. Frequently Asked Questions (FAQ)

Q1: What is circumferential stress in a rotating disc?
A: Circumferential stress (hoop stress) is the stress acting tangentially to 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, affecting the stress distribution.

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-like materials can have values close to 0.5.

Q4: Can this formula be used for hollow discs?
A: No, this specific formula is derived for solid discs. Hollow discs require different equations that account for inner and outer boundaries.

Q5: How does angular velocity affect the outer radius calculation?
A: Higher angular velocities increase centrifugal forces, requiring larger radii to maintain the same circumferential stress level, following an inverse square relationship.