Equations of Rotational Motion: Videos & Practice Problems

In rotational motion, four key equations parallel those in linear motion, using different variables: ωf = ωi + αt, ωf^2 = ωi^2 + 2αΔθ , Δθ = ωit + (1/2)αt^2, and Δθ = (1/2)(ωi + ωf)t . Understanding these equations is crucial for solving problems involving angular velocity and acceleration, such as determining the rotation in degrees or time taken for a wheel to reach a certain speed.

Equations of Rotational Motion

In the study of rotational motion, similar principles apply as in linear motion, but with different variables. Just as linear motion utilizes four key equations, rotational motion has its own set of four equivalent equations, which incorporate angular quantities. These equations are essential for solving problems involving rotational dynamics, where you may encounter variables such as angular velocity, angular acceleration, and angular displacement.

The primary equations for rotational motion are as follows:

1. \(\omega_f = \omega_i + \alpha t\)

2. \(\omega_f^2 = \omega_i^2 + 2\alpha \Delta \theta\)

3. \(\Delta \theta = \omega_i t + \frac{1}{2} \alpha t^2\)

4. \(\Delta \theta = \frac{1}{2} (\omega_i + \omega_f) t\)

In these equations, \(\omega\) represents angular velocity, \(\alpha\) is angular acceleration, and \(\Delta \theta\) denotes angular displacement. When solving problems, it is crucial to identify three known variables out of five possible ones, with one being the target variable and another being ignored. This approach helps determine which equation to use.

For example, consider a wheel that starts from rest (initial angular velocity \(\omega_i = 0\)) and accelerates at a constant rate of \(4 \, \text{radians/second}^2\) until it reaches a final angular velocity of \(80 \, \text{radians/second}\). To find the total angular displacement (\(\Delta \theta\)) in degrees, the appropriate equation is:

\(\Delta \theta = \frac{\omega_f^2 - \omega_i^2}{2\alpha}\)

Substituting the known values gives:

\(\Delta \theta = \frac{80^2 - 0^2}{2 \times 4} = 800 \, \text{radians}\)

To convert radians to degrees, use the conversion factor \(\frac{180 \, \text{degrees}}{\pi \, \text{radians}}\), resulting in:

\(800 \times \frac{180}{\pi} \approx 45,800 \, \text{degrees}\)

Next, to determine the time taken (\(\Delta t\)), you can use any equation that includes \(\Delta t\). The simplest choice is:

\(t = \frac{\Delta \theta - \omega_i t}{\alpha}\)

Rearranging and substituting the known values yields:

\(t = \frac{800}{4} = 20 \, \text{seconds}\)

In summary, while the equations for rotational motion mirror those of linear motion, they require careful attention to the specific variables and units involved. Understanding these concepts allows for effective problem-solving in rotational dynamics.

Rotational velocity of disc

In this scenario, we analyze a heavy disc with a radius of 20 meters that takes 1 hour to complete a full revolution. The period (T) of the disc is 1 hour, which converts to 3600 seconds. The disc accelerates from rest at a constant angular acceleration (α), and we aim to determine its final angular velocity (ω) after 1 hour.

To begin, we recognize that the disc completes one full revolution in this time, meaning the angular displacement (Δθ) after 1 hour is \(2\pi\) radians. This is crucial as it allows us to establish a relationship between the known variables and the unknown angular acceleration.

We utilize the third equation of motion for rotational dynamics, which is expressed as:

\[ \Delta \theta = \omega_{\text{initial}} t + \frac{1}{2} \alpha t^2 \]

Given that the initial angular velocity (\( \omega_{\text{initial}} \)) is 0 (the disc starts from rest), the equation simplifies to:

\[ \Delta \theta = \frac{1}{2} \alpha t^2 \]

Substituting the known values, we have:

\[ 2\pi = \frac{1}{2} \alpha (3600)^2 \]

Rearranging this equation to solve for α gives:

\[ \alpha = \frac{2 \cdot 2\pi}{(3600)^2} \approx 9.7 \times 10^{-7} \, \text{radians/second}^2 \]

Now that we have the angular acceleration, we can find the final angular velocity using the first equation of motion for rotation:

\[ \omega_{\text{final}} = \omega_{\text{initial}} + \alpha t \]

Substituting the known values, we find:

\[ \omega_{\text{final}} = 0 + (9.7 \times 10^{-7}) \cdot 3600 \approx 3.5 \times 10^{-3} \, \text{radians/second} \]

This result indicates that after 1 hour of constant acceleration, the disc achieves a final angular velocity of approximately \(3.5 \times 10^{-3}\) radians per second. The slow acceleration reflects the long period required for a complete revolution, emphasizing the relationship between time, angular displacement, and angular acceleration in rotational motion.

A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s². Calculate its angular speed 8 s after starting to accelerate.
BONUS:Calculate its linear displacement in the 8 s.

• A. 19 rad/s
• B. 29 rad/s
• C. 43 rad/s
• D. 53 rad/s

Answer: (B)

The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s² and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning? 
BONUS:How long (in seconds) does the turntable take to stop?

• A. 33.6 RPM
• B. 321 RPM
• C. 2,016 RPM
• D. 3,167 RPM

Answer: (B)

Here’s what students ask on this topic:

What are the equations of rotational motion?

The equations of rotational motion are analogous to those in linear motion but use different variables. They are:

${\omega }_f={\omega }_i+\alpha t$

${\omega }_{f}2={\omega }_{i}2+\; 2\alpha \Delta \theta$

$\Delta \theta ={\omega }_{i}t+\; (1/2)\alpha t2$

$\Delta \theta =\; (1/2)({\omega }_i+{\omega }_f)t$

These equations help solve problems involving angular velocity (ω), angular acceleration (α), and angular displacement (Δθ).

How do you convert angular displacement from radians to degrees?

To convert angular displacement from radians to degrees, use the conversion factor:

For example, if you have an angular displacement of 2 radians, the conversion to degrees is:

This conversion is essential when the problem requires the answer in degrees instead of radians.

What is the relationship between angular velocity and angular acceleration?

Angular velocity (ω) and angular acceleration (α) are related through the equation:

${\omega }_f={\omega }_i+\alpha t$

Here, ωf is the final angular velocity, ωi is the initial angular velocity, α is the angular acceleration, and t is the time. This equation shows that angular velocity changes linearly with time when angular acceleration is constant.

How do you determine the time taken for a wheel to reach a certain angular velocity?

To determine the time taken for a wheel to reach a certain angular velocity, use the equation:

Here, ωf is the final angular velocity, ωi is the initial angular velocity, and α is the angular acceleration. For example, if a wheel starts from rest (ω_i = 0) and reaches an angular velocity of 80 rad/s with an angular acceleration of 4 rad/s², the time taken is:

How do you solve for angular displacement using rotational motion equations?

To solve for angular displacement (Δθ), you can use one of the rotational motion equations. For example:

$\Delta \theta ={\omega }_{i}t+\; (1/2)\alpha t2$

Here, ωi is the initial angular velocity, α is the angular acceleration, and t is the time. If a wheel starts from rest (ω_i = 0) and accelerates at 4 rad/s² for 20 seconds, the angular displacement is:

$\Delta \theta =\; 0\; \times \; 20\; +\; (1/2)\; \times \; 4\; \times \; 202=\; 800\; ext\{\; radians\}$