Rotational Position & Displacement: Videos & Practice Problems

Rotational motion involves movement around a central point, described using angular position (θ) instead of linear position (x). Key concepts include angular displacement (Δθ), linked to linear displacement (Δx) through the equation ${\mathrm{\Delta x}}^{=}r\Delta \theta$, where r is the radial distance. A full revolution equals 360 degrees or 2π radians. Understanding these relationships is crucial for analyzing motion in circular paths, emphasizing the fixed nature of rotational origin and direction.

Rotational Position & Displacement

Rotational motion, also known as rotational kinematics, involves the movement of objects around a central point, forming circular paths. This concept can be contrasted with linear motion, where position is described using a variable \( x \). In rotational motion, we use the variable \( \theta \) to represent rotational position, which is measured in angles, typically in radians. Radians are a unit of angular measurement where one radian is approximately 57 degrees, and the relationship between degrees and radians is defined by the equation \( 360^\circ = 2\pi \) radians.

In linear motion, position is defined as the distance from the origin, measured in meters. In rotational motion, the rotational position is similarly defined as the distance from the origin, but it is measured in angles. The origin in linear motion can be arbitrary, meaning it can be set at any point, while in rotational motion, the origin is fixed at the positive x-axis, where \( \theta = 0 \). This distinction is crucial as it affects how we interpret motion in different contexts.

Directionality also differs between the two types of motion. In linear motion, the direction can be chosen arbitrarily, while in rotational motion, clockwise rotation is considered negative and counterclockwise rotation is positive. This fixed directionality is important for consistency in calculations and understanding motion.

Displacement in linear motion is represented as \( \Delta x \), while in rotational motion, it is represented as \( \Delta \theta \). The relationship between linear displacement and rotational displacement can be expressed with the equation:

\( \Delta x = r \Delta \theta \)

where \( r \) is the radial distance from the center of rotation. This equation highlights how linear and angular displacements are interconnected, allowing for conversions between the two.

When using this equation, it is essential to ensure that \( \Delta \theta \) is expressed in radians for the equation to yield accurate results. For example, if an object moves along a circular path with a radius of 10 meters and undergoes an angular displacement of 90 degrees, we first convert 90 degrees to radians:

\( 90^\circ = \frac{\pi}{2} \text{ radians} \)

Substituting this value into the equation gives:

\( \Delta x = 10 \times \frac{\pi}{2} \approx 15.7 \text{ meters} \)

This demonstrates how to calculate linear displacement from angular displacement, reinforcing the connection between the two forms of motion.

Displacement in Multiple Revolutions

Rotational displacement is a key concept in understanding motion around a circular path. When an object completes a full revolution, it undergoes a change in angle of 360 degrees, which is equivalent to \(2\pi\) radians. This relationship can be expressed using the equation for linear displacement, which is given by:

\( \Delta x = r \Delta \theta \)

Here, \(r\) represents the radius of the circle, and \(\Delta \theta\) is the change in angle. For one complete revolution, substituting \(\Delta \theta\) with \(2\pi\) leads to the formula for the circumference of a circle:

\( \Delta x = 2\pi r \)

This equation indicates that the linear distance traveled around the circle is equal to the circumference. If an object makes multiple revolutions, the total angular displacement can be calculated as:

\( \Delta \theta = 360^\circ \times n \quad \text{or} \quad \Delta \theta = 2\pi \times n \)

where \(n\) is the number of revolutions. Consequently, the linear displacement for \(n\) revolutions becomes:

\( \Delta x = 2\pi r \times n \)

To determine the number of revolutions from a given angle, simply divide the angle by 360 degrees or \(2\pi\) radians. For instance, if an object rotates through 720 degrees, dividing by 360 yields 2 revolutions. If the angle exceeds 360 degrees, subtracting 360 repeatedly will yield the equivalent angle within one revolution. For example, for an angle of 410 degrees, subtracting 360 results in 50 degrees, indicating the final position relative to the starting point.

In a practical example, consider an object that makes 2.2 revolutions around a circular path with a radius of 20 meters. The total angular displacement in degrees can be calculated as:

\( \Delta \theta = 360^\circ \times 2.2 = 792^\circ \)

To find the position relative to the starting point, subtract 360 degrees until the result is less than 360, leading to:

First subtraction: \(792 - 360 = 432\)

Second subtraction: \(432 - 360 = 72\)

Thus, the final position is 72 degrees from the starting point. For the linear displacement, it is essential to convert the angular displacement to radians:

\( \Delta \theta = 2\pi \times 2.2 \)

Calculating this gives:

\( \Delta x = r \Delta \theta = 20 \times (2\pi \times 2.2) \approx 276 \text{ meters} \)

This example illustrates the relationship between angular and linear displacement, emphasizing the importance of unit conversion and the application of rotational motion principles.

While you drive, your tires, all of radius 0.40 m, rotate 10,000 times. How far did you drive, in meters?

• A. 5027 m
• B. 8667 m
• C. 12566 m
• D. 25132 m

Answer: (D)

An object moves a total distance of 1,000 m around a circle of radius 30 m. How many degrees does the object go through? 
BONUS:How many complete revolutions does it make?

• A. 33.3°
• B. 488°
• C. 987°
• D. 1908°

Answer: (D)

A car travels a total of 2,000 m and 1140° around a circular path, starting from 0° . What is the radius of the circular path? 
BONUS:How far (in degrees) from 0° does the car end up?

• A. 1.754 m
• B. 100.5 m
• C. 780.8 m
• D. 2000 m

Answer: (B)

Here’s what students ask on this topic:

What is rotational motion and how is it different from linear motion?

Rotational motion involves movement around a central point, described using angular position (θ) instead of linear position (x). In linear motion, position is measured in meters, while in rotational motion, it is measured in angles (radians or degrees). The key difference is that in rotational motion, the origin and direction are fixed: the origin is always at the positive x-axis, and direction is defined as clockwise (negative) or counterclockwise (positive). In contrast, in linear motion, the origin and direction can be arbitrary.

How do you calculate angular displacement in rotational motion?

Angular displacement (Δθ) in rotational motion is calculated as the difference between the final and initial angular positions. The formula is:

$\mathrm{\Delta \theta }=\theta f_{}-\theta i_{}$

where θ_f is the final angular position and θ_i is the initial angular position. Angular displacement is typically measured in radians or degrees.

What is the relationship between linear displacement and angular displacement?

The relationship between linear displacement (Δx) and angular displacement (Δθ) is given by the equation:

$\mathrm{\Delta x}=r\mathrm{\Delta \theta }$

where r is the radial distance from the center of the circular path. This equation shows that linear displacement is directly proportional to the angular displacement and the radius of the circle.

How do you convert between degrees and radians in rotational motion?

To convert between degrees and radians, you can use the following relationships:

1 radian ≈ 57.3 degrees

360 degrees = 2π radians

To convert degrees to radians, use:

$\theta \left(\mathrm{rad}\right)=\theta \left(\mathrm{deg}\right)\times \frac{\pi }{180}$

To convert radians to degrees, use:

$\theta \left(\mathrm{deg}\right)=\theta \left(\mathrm{rad}\right)\times \frac{180}{\pi }$

What is the significance of the unit circle in rotational motion?

The unit circle is significant in rotational motion because it provides a standard reference for measuring angles. In the unit circle, the origin is fixed at the positive x-axis, and angles are measured counterclockwise from this point. This standardization helps in defining angular position and displacement consistently. The unit circle also explains why counterclockwise rotation is considered positive and clockwise rotation is negative, as it follows the mathematical convention of angle measurement.