Density Of Material Given Maximum Circumferential Stress In Solid Disc Calculator

Formula Used:

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

Circumferential Stress (σc):

Pascal

Angular Velocity (ω):

Radian per Second

Poisson's Ratio (ν):

(dimensionless)

Outer Radius Disc (router):

Meter

Density Of Disc (ρ):

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1. What is the Density Calculation Formula?

The formula calculates the density of material in a solid disc based on maximum circumferential stress, angular velocity, Poisson's ratio, and outer radius. This is particularly useful in mechanical engineering and material science for analyzing rotating disc systems.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula relates the material density to the stress distribution in a rotating solid disc, accounting for the disc's mechanical properties and rotational characteristics.

3. Importance of Density Calculation

Details: Accurate density calculation is crucial for designing rotating machinery components, predicting stress distributions, ensuring structural integrity, and selecting appropriate materials for specific applications.

4. Using the Calculator

Tips: Enter circumferential stress in Pascals, angular velocity in radians per second, Poisson's ratio (typically between 0.1-0.5), and outer radius in meters. All values must be positive and valid.

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 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 density values for disc materials?
A: Density varies by material: aluminum alloys (~2700 kg/m³), steel (~7850 kg/m³), titanium (~4500 kg/m³), depending on specific alloy composition.

Q4: When is this formula most applicable?
A: This formula is particularly useful for solid discs with uniform thickness rotating at constant angular velocity under plane stress conditions.

Q5: Are there limitations to this equation?
A: The formula assumes homogeneous, isotropic material properties and may not account for temperature variations, complex geometries, or non-uniform stress distributions.