You can find the time period of rotation (T), which is the time it takes for one complete rotation, primarily by using the object's angular velocity ($\omega$). The fundamental relationship is that the time period is equal to the total angular displacement of one full rotation (which is $2\pi$ radians) divided by the angular velocity.
Understanding Time Period and Angular Velocity
Before diving into calculations, let's quickly define the key terms:
- Time Period (T): The duration required for an object to complete one full revolution or rotation. It is typically measured in seconds.
- Angular Velocity ($\omega$): The rate at which an object rotates or revolves around an axis, measured in radians per second (rad/s) or sometimes degrees per second.
Using Angular Velocity ($\omega$)
The most common method for finding the time period of rotation relies on knowing the angular velocity. The formula that connects these two quantities is:
T = 2π / ω
Here:
Tis the time period of rotation.2πrepresents one full rotation in radians (approximately 6.283 radians).ω(omega) is the angular velocity in radians per second.
This formula essentially tells you how many seconds (T) are needed to cover the 2π radians of a full circle when you are rotating at a rate of ω radians per second.
Example from Reference Material
The provided reference material illustrates the calculation of a time period related to rotation, specifically the time period of Earth's rotation required for objects at the equator to become weightless. While the reference states "T=2πω′", the calculation performed uses 2π divided by the given value, which aligns with the standard formula T = 2π / ω'.
Let's look at the example calculation provided:
- The reference gives an angular velocity value (denoted as $\omega'$ in the reference's calculation context) of approximately $23.7 \times 10^{-5}$ s$^{-1}$ (this value corresponds to the angular velocity needed for equatorial weightlessness, derived from $g$ and Earth's radius).
- The calculation shown is:
T = 2π / (23.7 × 10⁻⁵ s⁻¹) - Performing this calculation gives:
T ≈ 5079.4 seconds - This result is then converted to minutes:
T ≈ 84.66 minutes
So, based on the calculation shown in the reference, the time period of rotation is found by dividing 2π by the angular velocity value $23.7 \times 10^{-5}$ s$^{-1}$. The reference provides the context: "The time period of rotation of the earth around its axis so that the objects at the equator become weightless is nearly (g=9.8m/s² , Radius of earth = 6400km.)".
Other Methods
For objects with slow, observable rotation, you could theoretically find the time period by directly measuring the time it takes for one complete rotation. However, for fast rotations or in theoretical calculations, using the angular velocity is the standard approach.
Key Terms Summary
| Term | Symbol | Definition | Calculation Relation to T |
|---|---|---|---|
| Time Period | T | Time for one complete rotation | - |
| Angular Velocity | ω or ω' | Rate of rotation (radians per second) | T = 2π / ω |
In summary, the most common and standard way to find the time period of rotation is by using the object's angular velocity and the formula T = 2π / ω.
