Angular Velocity Of Disc Given Constant At Boundary Condition For Circular Disc Calculator

Angular Velocity Formula:

\[ \omega = \sqrt{\frac{8 \times C_1}{\rho \times (r_{outer}^2) \times (3 + \nu)}} \]

Constant at boundary condition (C₁):

N/m²

Density Of Disc (ρ):

kg/m³

Outer Radius Disc (rₒᵤₜₑᵣ):

Poisson's Ratio (ν):

Angular Velocity (ω):

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1. What is Angular Velocity of Disc?

Angular velocity of a disc refers to how fast the disc rotates around its axis. It's a measure of the rate of change of angular displacement and is typically measured in radians per second.

2. How Does the Calculator Work?

The calculator uses the angular velocity formula:

\[ \omega = \sqrt{\frac{8 \times C_1}{\rho \times (r_{outer}^2) \times (3 + \nu)}} \]

Where:

\( \omega \) — Angular velocity (rad/s)
\( C_1 \) — Constant at boundary condition (N/m²)
\( \rho \) — Density of disc (kg/m³)
\( r_{outer} \) — Outer radius of disc (m)
\( \nu \) — Poisson's ratio (-)

Explanation: This formula calculates the angular velocity of a circular disc based on material properties and boundary conditions, considering the stress distribution in the disc.

3. Importance of Angular Velocity Calculation

Details: Calculating angular velocity is crucial for designing rotating machinery, analyzing stress distributions in rotating discs, and ensuring structural integrity in mechanical systems involving rotational motion.

4. Using the Calculator

Tips: Enter constant at boundary condition in N/m², density in kg/m³, outer radius in meters, and Poisson's ratio (typically between 0.1-0.5). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the boundary condition constant?
A: The constant at boundary condition represents the stress value obtained from the specific boundary conditions applied to the solid disc, which influences the angular velocity calculation.

Q2: How does Poisson's ratio affect angular velocity?
A: Poisson's ratio affects the stress distribution in the disc material. A higher Poisson's ratio typically results in a slightly lower angular velocity for the same boundary conditions.

Q3: What are typical angular velocity values for rotating discs?
A: Angular velocity values vary widely depending on application, from slow rotations (few rad/s) in large machinery to very high speeds (thousands of rad/s) in precision instruments.

Q4: Can this formula be used for non-circular discs?
A: No, this specific formula is derived for circular discs with specific boundary conditions. Different geometries require different formulations.

Q5: What are the limitations of this calculation?
A: This calculation assumes homogeneous material properties, perfect circular geometry, and specific boundary conditions. Real-world applications may require additional considerations.