Density Of Material Given Circumferential Stress At Center Of Solid Disc Calculator

Formula Used:

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

Circumferential Stress (σc):

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Outer Radius Disc (router):

Density Of Disc (ρ):

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1. What is the Density Calculation Formula?

The formula calculates the density of a solid disc material based on circumferential stress at the center, angular velocity, Poisson's ratio, and outer radius. It provides an accurate assessment of material density under rotational stress conditions.

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 (kg/m³)
\( \sigma_c \) — Circumferential stress (Pa)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio
\( r_{outer} \) — Outer radius of disc (m)

Explanation: The formula calculates material density by considering the stress distribution and rotational dynamics of a solid disc.

3. Importance of Density Calculation

Details: Accurate density calculation is crucial for material selection, structural analysis, and predicting the behavior of rotating components under stress conditions.

4. Using the Calculator

Tips: Enter circumferential stress in Pascals, angular velocity in rad/s, Poisson's ratio (typically 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?
A: Circumferential stress is the stress acting tangentially to the circumference of the disc, perpendicular to both the axis and radius of rotation.

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 engineering materials?
A: Density varies widely: aluminum ~2700 kg/m³, steel ~7850 kg/m³, titanium ~4500 kg/m³, depending on the specific alloy and composition.

Q4: When is this formula most applicable?
A: This formula is particularly useful for analyzing solid rotating discs in mechanical systems such as turbines, flywheels, and rotating machinery components.

Q5: Are there limitations to this equation?
A: The formula assumes homogeneous material properties, constant angular velocity, and may have reduced accuracy for very thin or very thick discs.