Understanding the Time Period of an Object with an Angular Velocity of 10 Radians per Second
Understanding the time period of an object moving with a given angular velocity is a fundamental concept in physics, particularly in oscillatory and rotational motion. In this article, we will explore the relationship between angular velocity and time period and how to compute the time period when given an angular velocity in radians per second.
Defining Angular Velocity and Time Period
Angular velocity is the rate of change of an angle, which is often given in radians per second (rad/s). It is denoted by ω (omega).
Time period (T) is the time taken for one complete cycle or revolution. It is the inverse of frequency (f) and is measured in seconds.
The relationship between angular velocity (ω) and frequency (f) is given by the formula:
f ω / (2π)
Converting Angular Velocity to Frequency
Given that the angular velocity (ω) is 10 rad/s, we first convert it to frequency (f):
f 10 / (2π) ≈ 1.59 Hz
The frequency is approximately 1.59 Hertz (Hz).
Using the relationship between frequency and time period:
T 1 / f
The time period is:
T 1 / (1.59) ≈ 0.628 seconds
Therefore, the time period of the object is approximately 0.628 seconds.
Interpreting the Results
The time period (T) is the time it takes for the object to complete one full revolution. In this case, when the angular velocity is 10 radians per second, the object takes about 0.628 seconds to complete one full revolution.
It is important to note that 10 radians per second is not a frequency but an angular velocity. The number of revolutions per second is calculated as follows:
1 revolution 2π radians
10 radians 5 / π revolutions per second
Therefore, the frequency (f) is 5/π revolutions per second or Hz.
Alternative Calculation
An alternative approach is to directly calculate the time period from the angular velocity:
Since 2π radians correspond to 1 revolution, 10 radians correspond to:
10 radians / (2π) 5 / π revolutions/second
The time period (T) is the reciprocal of the frequency:
T π / 5 seconds
In decimal form, the time period is 0.628318531 seconds, accurate to 9 significant figures.
Both methods confirm that the time period is approximately 0.628 seconds.