Density Of Material Given Circumferential Stress And Outer Radius Calculator

Formula Used:

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

Circumferential Stress (σc):

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Outer Radius Disc (router):

Radius of Element (r):

Density Of Disc (ρ):

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1. What is the Density Calculation Formula?

The formula calculates the density of a rotating disc material based on circumferential stress, angular velocity, Poisson's ratio, and geometric parameters. It's derived from the stress analysis of rotating discs in mechanical engineering.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula relates the material density to the stresses developed in a rotating disc, accounting for the disc's geometry and material properties.

3. Importance of Density Calculation

Details: Accurate density calculation is crucial for designing rotating machinery components, predicting stress distributions, and ensuring structural integrity under rotational forces.

4. Using the Calculator

Tips: Enter all values in appropriate units. Circumferential stress in Pascals, angular velocity in rad/s, Poisson's ratio (typically 0.1-0.5), radii 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 tensile stress acting tangentially to the circumference of the disc, caused by centrifugal forces during rotation.

Q2: Why does Poisson's ratio affect the density calculation?
A: Poisson's ratio represents the material's tendency to contract in directions perpendicular to the applied load, which influences how stresses distribute in the rotating disc.

Q3: What are typical density values for engineering materials?
A: Aluminum: ~2700 kg/m³, Steel: ~7850 kg/m³, Titanium: ~4500 kg/m³. Values vary based on specific alloys and material composition.

Q4: When is this formula most applicable?
A: This formula is particularly useful for analyzing constant-thickness rotating discs made of homogeneous, isotropic materials operating at constant angular velocity.

Q5: What are the limitations of this calculation?
A: The formula assumes ideal conditions and may not account for temperature effects, material anisotropy, variable thickness, or complex boundary conditions.