Linear Velocity Calculator
Calculate Linear Velocity
Linear Velocity vs. Radius at different Angular Velocities
What is Linear Velocity?
Linear velocity, also known as tangential velocity or tangential speed, is the rate at which an object moving along a circular path covers a distance along that path. Imagine a point on the edge of a spinning wheel; its linear velocity is how fast that point is moving along its circular trajectory at any given moment. It's a vector quantity, meaning it has both magnitude (speed) and direction (tangent to the circular path).
The Linear Velocity Calculator helps you determine this speed based on the object's angular velocity and its distance from the center of rotation (radius). Anyone studying physics, engineering, or dealing with rotating machinery can use this calculator. Common users include students, engineers designing rotating parts, and physicists analyzing circular motion.
A common misconception is confusing linear velocity with angular velocity. Angular velocity measures the rate of rotation (how quickly the angle changes), while linear velocity measures the straight-line speed of a point at a specific radius.
Linear Velocity Formula and Mathematical Explanation
The linear velocity (v) of a point on a rotating object is directly proportional to its angular velocity (ω) and its distance from the axis of rotation (radius, r). The formula is:
v = ω × r
Where:
- v is the linear velocity (tangential speed).
- ω (omega) is the angular velocity, which MUST be in radians per unit of time (e.g., radians per second) for this formula.
- r is the radius or distance from the axis of rotation.
If angular velocity is given in revolutions per minute (RPM) or other units, it must be converted to radians per second before using the formula. One revolution is equal to 2π radians, and one minute is 60 seconds.
So, ω (in rad/s) = RPM × (2π / 60).
Variables Table
| Variable | Meaning | Unit (SI) | Typical Range |
|---|---|---|---|
| v | Linear Velocity | meters per second (m/s) | 0 to thousands |
| ω | Angular Velocity | radians per second (rad/s) | 0 to thousands |
| r | Radius | meters (m) | 0 to hundreds |
| RPM | Revolutions Per Minute | rev/min | 0 to tens of thousands |
Practical Examples (Real-World Use Cases)
Example 1: Car Tire
A car tire with a radius of 30 cm (0.3 m) is rotating at 800 RPM. What is the linear velocity of a point on the outer edge of the tire?
- Convert RPM to rad/s: ω = 800 * (2π / 60) ≈ 83.78 rad/s
- Use the formula v = ω * r: v = 83.78 rad/s * 0.3 m ≈ 25.13 m/s
- Convert to km/h: 25.13 m/s * 3.6 ≈ 90.47 km/h
The point on the tire's edge is moving at approximately 25.13 m/s or 90.47 km/h relative to the axle.
Example 2: Wind Turbine Blade
The tip of a wind turbine blade is 50 meters from the center, and the turbine is rotating at 15 RPM. What is the linear velocity of the blade tip?
- Convert RPM to rad/s: ω = 15 * (2π / 60) ≈ 1.57 rad/s
- Use the formula v = ω * r: v = 1.57 rad/s * 50 m ≈ 78.5 m/s
- Convert to km/h: 78.5 m/s * 3.6 ≈ 282.6 km/h
The tip of the turbine blade travels at a very high speed of about 78.5 m/s or 282.6 km/h.
How to Use This Linear Velocity Calculator
- Enter Angular Velocity: Input the rate of rotation into the "Angular Velocity (ω)" field and select the correct units (RPM or rad/s) from the dropdown menu.
- Enter Radius: Input the distance from the center of rotation to the point of interest into the "Radius (r)" field and select its units (m, cm, in, ft).
- View Results: The calculator will automatically display the linear velocity in m/s, km/h, and mph, along with the intermediate values for angular velocity in rad/s and radius in meters. The formula used is also shown.
- Analyze Chart: The chart below the calculator visually represents how linear velocity changes with radius for different angular velocities, updating as you change the inputs.
- Reset or Copy: Use the "Reset" button to clear inputs to defaults or "Copy Results" to copy the calculated values.
This Linear Velocity Calculator provides instant results, helping you understand the relationship between rotational motion and linear speed.
Key Factors That Affect Linear Velocity Results
Several factors directly influence the calculated linear velocity:
- Angular Velocity (ω): The faster the object rotates (higher RPM or rad/s), the higher the linear velocity at any given radius. A doubling of angular velocity doubles the linear velocity.
- Radius (r): The further a point is from the axis of rotation (larger radius), the higher its linear velocity for the same angular velocity. A point twice as far from the center travels twice as fast linearly.
- Units of Angular Velocity: Ensuring the correct units (RPM or rad/s) are selected is crucial. The Linear Velocity Calculator handles the conversion, but incorrect unit selection will lead to wrong results.
- Units of Radius: Similarly, the units for radius (m, cm, in, ft) must be correctly specified for accurate conversion to meters within the calculation.
- Point of Interest: Linear velocity is specific to a point at a certain radius. Points at different radii on the same rotating object will have different linear velocities but the same angular velocity.
- Frame of Reference: The calculated linear velocity is relative to the axis of rotation. If the axis itself is moving, the overall velocity would be different.
Understanding these factors is key to correctly interpreting the results from the Linear Velocity Calculator.
Frequently Asked Questions (FAQ)
- Q1: What is the difference between linear velocity and angular velocity?
- A1: Angular velocity measures how fast an object rotates or spins around an axis (rate of change of angle, measured in rad/s or RPM). Linear velocity measures how fast a point at a certain radius on that object is traveling along its circular path (measured in m/s, km/h, etc.).
- Q2: Can two points on the same spinning disk have the same linear velocity?
- A2: Only if they are at the same distance (radius) from the center of rotation. Points at different radii will have different linear velocities but the same angular velocity.
- Q3: Why must angular velocity be in radians per second for the formula v = ω * r?
- A3: The formula is derived using radians as the measure of angle because the arc length (s) covered is s = θ * r, where θ is in radians. Velocity is the rate of change of distance (ds/dt = d(θ*r)/dt = r * dθ/dt = r * ω), so ω must be in rad/s for v to be in units of length per second.
- Q4: What if the object is not rotating uniformly?
- A4: If the angular velocity is changing (there is angular acceleration), the linear velocity at any instant is still given by v = ω * r, but ω will be the instantaneous angular velocity at that moment.
- Q5: Does the mass of the object affect its linear velocity?
- A5: The linear velocity of a point on a rotating rigid body depends only on its angular velocity and radius, not directly on the mass of the object itself. However, the force required to achieve that rotation depends on the mass distribution (moment of inertia).
- Q6: How does this calculator handle different units?
- A6: The Linear Velocity Calculator converts the input angular velocity to rad/s and the input radius to meters before applying the formula v = ω * r. The result is then displayed in m/s and also converted to km/h and mph.
- Q7: What is tangential velocity?
- A7: Tangential velocity is another term for linear velocity when referring to circular motion. It's called "tangential" because the velocity vector is always tangent to the circular path at that point.
- Q8: Can I use this calculator for objects rolling without slipping?
- A8: Yes. For an object rolling without slipping (like a wheel on the ground), the linear velocity of the center of mass of the wheel relative to the ground is equal to ω * r, where r is the radius of the wheel and ω is its angular velocity.
Related Tools and Internal Resources
Explore other calculators and resources that might be helpful:
- Angular Velocity Calculator: Calculate angular velocity from frequency or period.
- Centripetal Force Calculator: Determine the force required to keep an object in circular motion.
- Rotational Kinetic Energy Calculator: Calculate the energy of a rotating object.
- Speed, Distance, Time Calculator: Basic calculator for linear motion.
- Torque Calculator: Calculate torque based on force and distance.
- More Physics Calculators: A collection of various physics-related calculators.