Question

The angular acceleration for an object undergoing circular motion is plotted versus time in the figure. If the object started from rest at \(t=0\), the net angular displacement of the object at \(t=t_{6}\) a) is in the clockwise direction. b) is in the counterclockwise direction. c) is zero. d) cannot be determined

Short answer

Answer: d) Cannot be determined.

Step-by-step solution

Finding angular velocity

To find the angular velocity, we need to integrate the given angular acceleration function with respect to time. Since we're not given a specific function, we'll describe this step in general terms. Let \(\alpha(t)\) be the given angular acceleration function of time, \(t\). Then, the angular velocity, \(\omega(t)\), can be found using the following integral: $$\omega(t) = \int \alpha(t) dt.$$ Find the value of this integral between \(t=0\) and \(t=t_\mathrm{f}\). We are not given a specific function, so we will keep this step general.

Finding angular displacement

Now that we have found the angular velocity function, we can find the angular displacement by integrating the angular velocity function with respect to time. Let \(\omega(t)\) be the angular velocity function we found in Step 1. Then, the angular displacement, \(\theta(t)\), can be found using the following integral: $$\theta(t) = \int \omega(t) dt.$$ Find the value of this integral between \(t=0\) and \(t=t_\mathrm{f}\). Again, we'll keep this step general since we don't have a specific function.

Determine the direction of angular displacement

With the angular displacement function \(\theta(t)\) found in Step 2, we can determine the net angular displacement at \(t=t_\mathrm{f}\). If \(\theta(t_\mathrm{f}) > 0\), then the net angular displacement is in the counterclockwise direction. If \(\theta(t_\mathrm{f}) < 0\), then the net angular displacement is in the clockwise direction. If \(\theta(t_\mathrm{f}) = 0\), then the net angular displacement is zero. Given the information provided, we cannot determine the exact direction since we do not have specific values or a plot. Therefore, the correct answer is: d) Cannot be determined.

Key concepts

Angular Acceleration

Angular acceleration is a key concept when examining objects in circular motion. It measures the rate at which an object's rotational speed, or angular velocity, changes over time. In mathematical terms, angular acceleration, denoted as \( \alpha(t) \), is the derivative of the angular velocity \( \omega(t) \) with respect to time. This relationship is represented by the formula
\( \alpha(t) = \frac{d\omega(t)}{dt} \).

If you think of it as the rotational equivalent of linear acceleration, it becomes easier to understand. Just as a car might accelerate from a stoplight by increasing its speed, an object in circular motion might start spinning faster, which would be a sign of positive angular acceleration. Similarly, if the object starts to spin more slowly, it would be experiencing negative angular acceleration. It's crucial to know the direction and magnitude of angular acceleration to predict an object's rotational motion over time.

Circular Motion

Circular motion occurs when an object moves along a circular path. In physics, this type of motion is key to understanding how bodies rotate, such as planets orbiting a star or a carousel turning at the playground. Two main types of circular motion are uniform circular motion, where speed is constant, and non-uniform circular motion, where the speed changes.

During circular motion, the object has an angular velocity \( \omega \), which describes how fast it is rotating around the circle's center. However, the concept of angular displacement, denoted as \( \theta \), is just as crucial. It represents the angle through which the object has rotated. Think of angular displacement as the rotational distance from the starting point. If an object completes a full rotation, it has an angular displacement of \( 2\pi \) radians, equivalent to 360 degrees. Circular motion is deeply connected to the concepts of angular velocity and acceleration, and understanding these relationships is fundamental to mastering physics problems involving rotation.

Integrating Angular Velocity

Integrating angular velocity is the process of determining the angular displacement of an object that's in motion. Since angular velocity, \( \omega(t) \), is the rate of change of angular displacement over time, integrating it with respect to time gives us the total angular displacement, \( \theta(t) \), over a given period.

For example, if we want to know how much an object has rotated between time \( t=0 \) and some time \( t=t_{f} \), we would compute the integral of angular velocity over this interval:
\[ \theta(t_f) = \int_0^{t_f} \omega(t) \, dt \].
The result of this integration will tell us the object's angular position relative to its starting point. If the result is positive, the object has moved in the counterclockwise direction, while a negative result indicates clockwise movement. The integral also takes into account any variations in speed during the movement, making it a powerful tool for understanding complex rotational behavior.