Sums on Angular Displacement When Angular Velocity is Constant

Angular displacement, angular velocity, and angular acceleration are key concepts in the study of rotational motion in physics. When an object rotates about an axis, it undergoes angular displacement, which is the angle through which it has rotated. If the angular velocity is constant, the motion is uniform, and the calculations become more straightforward.

Understanding Angular Displacement

Angular displacement ($\theta$) is measured in radians (rad), degrees, or revolutions. It represents the angle through which a point or line has been rotated in a specified sense about a specified axis. One complete revolution is equal to $2\pi$ radians or 360 degrees.

Angular Velocity and Its Constancy

Angular velocity ($\omega$) is the rate of change of angular displacement and is measured in radians per second (rad/s). When an object rotates with constant angular velocity, there is no angular acceleration ($\alpha$), meaning that $\alpha = 0$.

Relationship Between Angular Displacement and Angular Velocity

When the angular velocity is constant, the angular displacement can be calculated using the formula:

[ \theta = \omega t ]

where:

• $\theta$ is the angular displacement in radians,
• $\omega$ is the constant angular velocity in rad/s,
• $t$ is the time in seconds for which the object has been rotating.

Table: Key Differences and Important Points

Aspect	Description
Angular Displacement	The angle through which an object has rotated about a fixed axis.
Angular Velocity	The rate at which the object rotates, measured in radians per second.
Constant Angular Velocity	Implies no change in the rate of rotation, hence no angular acceleration.
Time	The duration for which the object has been in rotational motion.
Formula	$\theta = \omega t$ when $\omega$ is constant.

Examples

Example 1: Calculating Angular Displacement

A wheel rotates with a constant angular velocity of 120 rad/s. Calculate the angular displacement of the wheel after 10 seconds.

Using the formula $\theta = \omega t$, we have:

[ \theta = 120 \, \text{rad/s} \times 10 \, \text{s} = 1200 \, \text{rad} ]

Example 2: Finding Time from Angular Displacement and Angular Velocity

A disc makes 5 complete revolutions with a constant angular velocity of 50 rad/s. How long does it take for the disc to complete these revolutions?

First, convert revolutions to radians:

[ 5 \, \text{revolutions} \times \frac{2\pi \, \text{rad}}{1 \, \text{revolution}} = 10\pi \, \text{rad} ]

Now, use the formula $\theta = \omega t$ to find $t$:

[ 10\pi \, \text{rad} = 50 \, \text{rad/s} \times t ]

[ t = \frac{10\pi \, \text{rad}}{50 \, \text{rad/s}} = \frac{\pi}{5} \, \text{s} \approx 0.628 \, \text{s} ]

Example 3: Angular Displacement in Degrees

A fan blade rotates at a constant angular velocity of 300 rad/s. Calculate the angular displacement in degrees after 2 seconds.

First, calculate the angular displacement in radians:

[ \theta = 300 \, \text{rad/s} \times 2 \, \text{s} = 600 \, \text{rad} ]

Now, convert radians to degrees:

[ 600 \, \text{rad} \times \frac{180^\circ}{\pi \, \text{rad}} \approx 34377.5^\circ ]

Conclusion

When dealing with sums on angular displacement with constant angular velocity, the key is to remember the direct proportionality between angular displacement and time, as expressed by the formula $\theta = \omega t$. This relationship simplifies calculations and is fundamental in solving problems related to uniform rotational motion.