Calculating Rotational Motion: Constant Angular Acceleration and Revolutions in 10 Seconds

Rotational motion is a fundamental concept in physics, with applications across various fields from aerospace engineering to robotics. In particular, understanding the behavior of objects under constant angular acceleration is crucial for a range of calculations. In this article, we will explore how to determine the number of revolutions a wheel makes when subjected to constant angular acceleration over a given time period.

Understanding Rotational Motion

Rotational motion can be described using several key concepts, including angular displacement, initial angular velocity, angular acceleration, and time. These concepts are analogous to the linear motion equations, with angular measurements replacing their linear counterparts.

Rotational Motion Equations

The equations governing rotational motion are similar to those used in linear motion. For rotational motion, these equations can be expressed as:

theta omega_0 t 0.5 alpha t^2

Where:

theta is the angular displacement in radians omega_0 is the initial angular velocity in rad/s alpha is the angular acceleration in rad/s2 t is the time in seconds

Problem Scenario

Let's consider a wheel that starts from rest and rotates with a constant angular acceleration. We want to determine the number of revolutions it makes in the first 10 seconds.

Given:

Initial angular velocity, omega_0 0 rad/s (the wheel starts from rest) Angular acceleration, alpha 2 rad/s2 Time, t 10 seconds

Solving for Angular Displacement

Using the rotational motion equation, we substitute the given values:

theta 0 cdot 10 0.5 cdot 2 cdot 10^2

Now, let's calculate the angular displacement:

theta 0 0.5 cdot 2 cdot 10^2 0 1 cdot 100 100 radians

Converting Radians to Revolutions

To convert radians to revolutions, we use the fact that 2π radians equal one revolution:

Revolutions frac{theta}{2pi} frac{100}{2pi} approx frac{100}{6.2832} approx 15.92

Therefore, the wheel makes approximately 15.92 revolutions in the first 10 seconds.

Further Insights

The same kinematic equations that govern linear motion can be applied to rotational motion with the appropriate substitutions. Here, s (displacement) is replaced by theta (angular displacement), u (initial velocity) is replaced by omega_0 (initial angular velocity), and a (acceleration) is replaced by alpha (angular acceleration).

The second equation of motion in rotational motion can also be utilized:

s ut 0.5 at^2

Substituting the given values, we find that the angular distance traveled is 100 radians, which is approximately 15.9 revolutions (since 1 revolution 2π radians).

The wheel will complete 15 full revolutions in approximately 9.7 seconds and will be just past the 16th revolution at the 10-second mark.