Rotational Velocity & Acceleration: Videos & Practice Problems

In rotational motion, angular velocity (ω) is defined as the change in angular position (Δθ) over time (Δt), measured in radians per second. Angular acceleration (α) is the change in angular velocity (Δω) over time (Δt), measured in radians per second squared. The period (T) is the time for one complete revolution, while frequency (f) is the number of revolutions per second. The relationship between these variables is given by ω = 2πf and T = 1/f. RPM (revolutions per minute) can be converted to frequency using f = RPM/60.

Rotational Velocity & Acceleration

In the study of rotational motion, understanding the relationship between linear and angular quantities is essential. Just as linear position and displacement are represented by x and Δx, respectively, in rotational motion, these quantities are denoted by θ (theta) and Δθ. The concepts of velocity and acceleration also have their rotational counterparts: angular velocity and angular acceleration, represented by the Greek letters ω (omega) and α (alpha).

Average linear velocity is defined as the change in position over time, expressed mathematically as:

\[\bar{v} = \frac{\Delta x}{\Delta t}\]

with units of meters per second (m/s). In contrast, angular velocity is given by:

\[\omega = \frac{\Delta \theta}{\Delta t}\]

where ω is measured in radians per second (rad/s). Similarly, linear acceleration is defined as the change in velocity over time:

\[a = \frac{\Delta v}{\Delta t}\]

with units of meters per second squared (m/s²), while angular acceleration is expressed as:

\[\alpha = \frac{\Delta \omega}{\Delta t}\]

with units of radians per second squared (rad/s²).

To further understand angular motion, it is important to recognize the relationship between angular velocity, period, and frequency. The period T is the time taken for one complete revolution, and it can be related to angular velocity by the equation:

\[\omega = \frac{2\pi}{T}\]

Frequency f, which is the number of revolutions per second, is the inverse of the period:

\[f = \frac{1}{T}\]

Additionally, frequency can be expressed in terms of RPM (revolutions per minute) as:

\[f = \frac{RPM}{60}\]

These relationships allow for conversions between different measures of rotational motion. For instance, if an object spins at 120 RPM, converting this to frequency involves dividing by 60, yielding:

\[f = \frac{120}{60} = 2 \text{ Hz}\]

From frequency, one can find the period:

\[T = \frac{1}{f} = \frac{1}{2} = 0.5 \text{ seconds}\]

To find angular velocity ω, the equation:

\[\omega = 2\pi f\]

can be used. Substituting the frequency gives:

\[\omega = 2\pi(2) = 4\pi \approx 12.57 \text{ rad/s}\]

In summary, the study of rotational motion involves understanding how angular quantities relate to their linear counterparts, utilizing key equations to convert between different measures of motion, and recognizing the significance of point masses and rigid bodies in analyzing rotational dynamics.

Rotational velocity of Earth

Understanding the rotational velocity of Earth involves analyzing two distinct scenarios: its rotation around itself and its orbit around the Sun. The angular velocity, denoted as \( \omega \), can be calculated using the formula \( \omega = \frac{\Delta \theta}{\Delta t} \) or \( \omega = \frac{2\pi}{T} \) or \( \omega = 2\pi f \), where \( T \) is the period of rotation and \( f \) is the frequency.

For Earth's rotation around itself, the period \( T \) is 24 hours. To convert this into seconds, we calculate:

\( T = 24 \, \text{hours} \times 60 \, \text{minutes/hour} \times 60 \, \text{seconds/minute} = 86400 \, \text{seconds} \).

Substituting this into the angular velocity formula gives:

\( \omega = \frac{2\pi}{86400} \approx 7.27 \times 10^{-5} \, \text{radians/second} \).

This small value reflects the long duration of Earth's rotation relative to faster rotating objects.

Next, we consider Earth's rotation around the Sun, which has a period of 365 days. Again, we convert this into seconds:

\( T = 365 \, \text{days} \times 24 \, \text{hours/day} \times 60 \, \text{minutes/hour} \times 60 \, \text{seconds/minute} = 31,536,000 \, \text{seconds} \).

Using the angular velocity formula, we find:

\( \omega = \frac{2\pi}{31,536,000} \approx 1.99 \times 10^{-7} \, \text{radians/second} \).

This value is even smaller than that of Earth's rotation around itself, illustrating the vast difference in time scales between the two motions.

It is important to note the distinction between these two scenarios. When Earth rotates around itself, it can be approximated as a rigid body due to its spherical shape. In contrast, when it orbits the Sun, it behaves more like a point mass moving along a circular path, given the relative sizes and distances involved. This conceptual understanding is crucial for analyzing rotational dynamics in different contexts.

Calculate the rotational velocity (in rad/s) of a clock's minute hand. 
EXTRA:Calculate the rotational velocity (in rad/s) of a clock's hour hand.

• A. 8.7×10⁻⁴ rad/s
• B. 1.7×10⁻³ rad/s
• C. 0.0167 rad/s
• D. 0.104 rad/s

Answer: (B)

A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. Calculate its angular acceleration.

• A. 6.28 rad/s²
• B. 12.6 rad/s²
• C. 18.8 rad/s²
• D. 377 rad/s²

Answer: (A)

Here’s what students ask on this topic:

What is the difference between angular velocity and angular acceleration?

Angular velocity (ω) measures how quickly an object rotates or spins around a central point, defined as the change in angular position (Δθ) over time (Δt). It is measured in radians per second (rad/s). The equation is:

$\omega =\frac{\mathrm{\Delta \theta }}{\mathrm{\Delta t}}$

Angular acceleration (α) measures how quickly the angular velocity changes over time. It is defined as the change in angular velocity (Δω) over time (Δt) and is measured in radians per second squared (rad/s²). The equation is:

$\alpha =\frac{\mathrm{\Delta \omega }}{\mathrm{\Delta t}}$

How do you convert RPM to angular velocity?

To convert RPM (revolutions per minute) to angular velocity (ω) in radians per second, you can use the following steps:

1. Convert RPM to frequency (f) in hertz (Hz) using the equation:

$f=\frac{\mathrm{RPM}}{60}$

2. Convert frequency to angular velocity using the equation:

$\omega =2\pi f$

For example, if an object spins at 120 RPM, first convert to frequency:

$f=\frac{120}{60}=2$

Then convert to angular velocity:

$\omega =2\pi \left(2\right)=4\pi \mathrm{rad}/s$

What is the relationship between period and frequency in rotational motion?

The period (T) and frequency (f) in rotational motion are inversely related. The period is the time it takes for one complete revolution, measured in seconds, while frequency is the number of revolutions per second, measured in hertz (Hz). The relationship is given by:

$T=\frac{1}{f}$

Conversely, frequency can be expressed as:

$f=\frac{1}{T}$

For example, if the period of a rotating object is 0.5 seconds, the frequency is:

$f=\frac{1}{0.5}=2 \mathrm{Hz}$

How do you calculate angular velocity from the period of rotation?

To calculate angular velocity (ω) from the period of rotation (T), you can use the relationship between angular velocity, period, and frequency. First, find the frequency (f) using the period:

$f=\frac{1}{T}$

Then, use the frequency to find the angular velocity:

$\omega =2\pi f$

Combining these equations, you get:

$\omega =\frac{2\pi }{T}$

For example, if the period of rotation is 2 seconds, the angular velocity is:

$\omega =\frac{2\pi }{2}=\pi \mathrm{rad}/s$

What are the units of angular velocity and angular acceleration?

Angular velocity (ω) is measured in radians per second (rad/s). It represents how quickly an object rotates or spins around a central point. The equation for angular velocity is:

$\omega =\frac{\mathrm{\Delta \theta }}{\mathrm{\Delta t}}$

Angular acceleration (α) is measured in radians per second squared (rad/s²). It represents how quickly the angular velocity changes over time. The equation for angular acceleration is:

$\alpha =\frac{\mathrm{\Delta \omega }}{\mathrm{\Delta t}}$

These units are essential for understanding rotational motion and are analogous to linear velocity (m/s) and linear acceleration (m/s²) in linear motion.