# How to Derive Period From Angular Velocity

## What is Period in Physics?

In physics, the term “period” refers to the time it takes for a repetitive or oscillatory motion to complete one full cycle. It is often denoted by the symbol “T.” The concept of period is fundamental in the study of waves, vibrations, and periodic motion. Now, let us see how to derive the formula for the period from angular velocity.

For example, in the context of a simple pendulum, the period is the time it takes for the pendulum bob to swing back and forth, returning to its starting position. In the case of a wave, such as a sine wave, the period is the time it takes for one complete wave cycle to pass a given point in space.

The period is related to the frequency (f) of the motion or wave through the equation:

T = 1 / f

Where:

• T is the period (measured in seconds).
• f is the frequency (measured in hertz, Hz), which represents the number of cycles or oscillations that occur per second.

## What is Angular Velocity?

Angular velocity is the ratio of the angle turned through to the elapsed time. We can also define angular velocity as the change in the angle per second. The unit of angular velocity is radians per second (rad/s or rads-1).

## Derivation of Period From Angular Velocity

From velocity, we know that velocity, v = displacement (s) / time (t)

v=s/t

and s is the displacement which represents the length of the arc. In a circle, we have an arc s, the circle of the radius r, and the angle θ.

Now, if we allow θ to represent v and plugin r instead of t, we will get

θ = s/r

When we cross multiply the above expression to find s, we will end up with

s = θr

We can now go ahead to divide the above equation by time t to obtain

s/t = θr/t

We can clearly see that s/t (which is the ratio of displacement to time) is defined as the velocity of an object (v). Thus, we can now define the above expression as

v = s/t = θr/t

v is the linear velocity of the body

The above expression is the same as

v = (θ/t) x r

But θ/t is equivalent to  ω (which is the angular velocity)

Hence,

v=ωr

The above formula is the relationship between linear velocity and the angular velocity in radians.

To arrive at the formula for period T, we can say that

T=2π/ω [where 2π represents the circumference of a circle which is 2 x 1800 = 3600]

But ω = 2πf

Which implies that

T = 2π/2πf = 1/f

Thus, the period is defined as

T = 1/f [where f is the frequency]

Hence, this is the method we apply to understand how to derive the formula for the period from angular velocity.

## The Basics: Angular Velocity and Period

### Angular Velocity Explained

Angular velocity is a fundamental concept in rotational motion. It measures how quickly an object rotates around an axis. To calculate angular velocity (ω), use the formula:

ω = Δθ / Δt

Where:

• ω is angular velocity (in radians per second)
• Δθ is the change in angle (in radians)
• Δt is the change in time (in seconds)

### Period Defined

Now, let’s talk about the period (T). Period is the time it takes for an object to complete one full rotation around an axis. It is inversely proportional to angular velocity and can be calculated using the formula:

T = 2π / ω

Where:

• T is the period (in seconds)
• ω is the angular velocity (in radians per second)

## How to Derive Period From Angular Velocity

To derive the period from angular velocity, follow these steps:

1. Obtain the Angular Velocity: Measure the angular velocity (ω) of the rotating object. Ensure your measurement is in radians per second.
2. Apply the Formula: Use the formula for calculating the period (T = 2π / ω) to derive the period. This formula relates the angular velocity to the period.
3. Calculate the Period: Plug in the angular velocity value you obtained into the formula and perform the calculation. The result will give you the period of rotation.
4. Interpret the Result: The calculated value represents the time it takes for one complete rotation. It’s your period!

### What are the units of angular velocity?

Angular velocity is typically measured in radians per second (rad/s). In some cases, degrees per second (°/s) may also be used, but radians per second are the standard SI units.

### Can angular velocity be negative?

Yes, angular velocity can be negative. A negative angular velocity indicates that the object is rotating in the opposite direction (counterclockwise) compared to a positive angular velocity (clockwise).

### Is angular velocity the same as linear velocity?

No, they are not the same. Angular velocity measures the rate of rotation, while linear velocity measures the rate of change of linear position. Linear velocity depends on the object’s radius and angular velocity.

### How can I measure angular velocity accurately?

You can measure angular velocity accurately using various tools, such as tachometers, encoders, or by analyzing position data over time and calculating the change in angle.

### What is the relationship between frequency and period?

Frequency (f) is the reciprocal of the period (T), meaning that: f = 1 / T As the period increases, the frequency decreases, and vice versa.

### How is angular velocity used in real-world applications?

Angular velocity plays a crucial role in various applications, including robotics, automotive engineering (e.g., measuring wheel rotation), and astronomy (e.g., calculating the rotation of celestial bodies).

## Conclusion

In this comprehensive guide, we’ve explored the concept of deriving the period from angular velocity. We’ve covered the fundamental equations, explained the relationship between these two parameters, and provided you with a step-by-step guide on how to calculate the period accurately. Whether you’re studying physics or applying this knowledge in engineering projects, mastering this concept is essential.

## Note

The relationship between angular velocity and period is very important in the field of physics and engineering. Whether you are a student looking to ace your physics class or an engineer working on complex projects, knowing how to derive the period from angular velocity is essential.

You may also like to read:

How to Calculate Angular Velocity in Radians per Second

Centripetal Acceleration: Definition, Formula, Unit, and Calculations

How to Calculate Centripetal Force

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