Problem 10 - College Physics

Question

At \(t=0\), a cooling fan running at \(200 \mathrm{rad} / \mathrm{s}\) is turned off and then slows down at a rate of \(20 \mathrm{rad} / \mathrm{s}^{2}\). Simultaneously (at \(t=0\) ), a second cooling fan is turned on and begins to spin from rest with an acceleration of \(60 \mathrm{rad} / \mathrm{s}^{2}\). (a) Find the time at which both fans have the same angular speed. (b) What is the angular speed of the fans at this time?

Step-by-Step Solution

Answer

Both fans reach the same angular speed of 150 rad/s at 2.5 seconds.

Step 1: Define the problem variables

Let \( \omega_1(t) \) be the angular speed of the first fan, and \( \omega_2(t) \) be the angular speed of the second fan at time \( t \). Initially, \( \omega_1(0) = 200 \mathrm{rad/s} \) and \( \omega_2(0) = 0 \mathrm{rad/s} \). The deceleration of the first fan is \( \alpha_1 = -20 \mathrm{rad}/\mathrm{s}^2 \), and the acceleration of the second fan is \( \alpha_2 = 60 \mathrm{rad}/\mathrm{s}^2 \).

Step 2: Write the equations of motion for each fan

For the first fan: \( \omega_1(t) = \omega_1(0) + \alpha_1 \cdot t = 200 - 20t \).For the second fan: \( \omega_2(t) = \omega_2(0) + \alpha_2 \cdot t = 0 + 60t \).

Step 3: Set the angular speeds equal to find the time

Set \( \omega_1(t) = \omega_2(t) \). \( 200 - 20t = 60t \)Combine like terms:\( 200 = 80t \)Solve for \( t \): \( t = \frac{200}{80} = 2.5 \text{ seconds} \).

Step 4: Find the angular speed at this time

Substitute \( t = 2.5 \) seconds into either angular speed equation:\( \omega_1(2.5) = 200 - 20 \times 2.5 \)\( \omega_1(2.5) = 200 - 50 \)\( \omega_1(2.5) = 150 \mathrm{rad/s} \).Since \( \omega_1(t) = \omega_2(t) \), \( \omega_2(2.5) = 150 \mathrm{rad/s} \) as well.

Key Concepts

Angular Motion

Angular motion refers to the rotation of an object around a point or axis. In our case of the cooling fans, each fan rotates about a central axis, creating angular motion. This motion is described using angular velocity, which is measured in radians per second (rad/s).
Angular velocity tells us how fast the fan is spinning. In problems like this, we often use the symbol \( \omega \) to represent angular speed.

• For the first fan, it starts at an initial velocity of 200 rad/s.
• The second fan starts from rest, meaning its initial angular velocity is 0 rad/s.

The change in speed over time is affected by two factors: acceleration and deceleration, which we will explore next.

Deceleration

Deceleration is simply the slowing down of an object. For angular motion, this means the object's rate of spin decreases over time.
In our exercise, the first fan undergoes deceleration, slowing at a rate of 20 rad/s². This deceleration acts in the opposite direction to its initial angular velocity, causing the fan to gradually slow down.
Deceleration affects how quickly or slowly an object stops spinning. It's described using a negative angular acceleration value because it reduces the angular speed of the object. The formula for finding the new angular speed at time \( t \) can be expressed as:
\[ \omega(t) = \omega(0) + \alpha \cdot t \] where \( \alpha \) (here \(-20\,\mathrm{rad/s}^2\)) is the deceleration rate.

Acceleration

Acceleration, in the context of angular motion, refers to how quickly an object's spin increases. It is characterized by a positive change in angular velocity.
In the exercise, the second fan accelerates from a standstill with an angular acceleration of 60 rad/s².
A positive angular acceleration means the fan starts spinning faster as time progresses. Just like deceleration, the angular speed at any given time can be calculated using the formula:
\[ \omega(t) = \omega(0) + \alpha \cdot t \] where here, \( \alpha \) is a positive 60 rad/s², reflecting the fan's increasing speed.

Equations of Motion

Equations of motion are mathematical expressions used to describe the motion of objects. They help predict future positions or speeds of moving objects based on known quantities like initial speed or acceleration.
In this problem, the equations of motion for angular velocity are applied to both fans:

• First fan: \( \omega_1(t) = 200 - 20t \)
• Second fan: \( \omega_2(t) = 60t \)

Setting these equations equal helps us find when both fans reach the same angular speed, which happens at 2.5 seconds. At this moment, both fans spin at 150 rad/s.
Understanding equations of motion lets us solve complex problems involving rotation and enables us to predict future states of rotating systems.