Question

Determine the linear and angular speeds and accelerations of a speck of dirt located \(2.0 \mathrm{~cm}\) from the center of a CD rotating inside a CD player at 250 rpm.

Short answer

Answer: The linear speed of the speck of dirt is approximately 0.5236 meters per second, and its linear acceleration is 0. The angular speed is approximately 26.18 radians per second, and the angular acceleration is 0.

Step-by-step solution

Convert rotational speed to angular speed

To find the angular speed of the speck of dirt on the CD, we first need to convert the given rotational speed (250 rpm) to angular speed in radians per second. To do this, we can use the conversion factor: 1 revolution = \(2\pi\) radians. Also, we need to convert minutes to seconds using the conversion factor: 1 minute = 60 seconds. The conversion from rpm to radians per second can be done as follows: Angular speed (in radians per second) = (Rotational speed (in rpm) * (2π radians per revolution)) / (60 seconds per minute)

Calculate the angular speed

Now, we can plug in the given values to calculate the angular speed: Angular speed = (250 * 2π) / 60 ≈ 26.18 radians per second

Calculate the linear speed

Next, we need to find the linear speed of the speck of dirt. Linear speed (v) can be calculated using the formula: v = rω, where r is the radius (distance from the center) and ω is the angular speed. Given that the speck of dirt is 2.0 cm from the center of the CD, we can convert the distance to meters (to keep units consistent) by dividing by 100. So, the radius is 0.02 meters. Now we can calculate the linear speed: Linear speed = rω = 0.02 * 26.18 ≈ 0.5236 meters per second

Calculate the angular acceleration

Since the CD is rotating at a constant speed (250 rpm), there is no angular acceleration acting on the speck of dirt. Therefore, the angular acceleration is 0.

Calculate the linear acceleration

Finally, we need to find the linear acceleration of the speck of dirt. Linear acceleration (a) can be calculated using the formula: a = rα, where r is the radius (distance from the center) and α is the angular acceleration. As we found in Step 4, the angular acceleration is 0. This means that the linear acceleration is also 0 since there is no change in the linear speed of the speck of dirt. To summarize, the linear speed of the speck of dirt is approximately 0.5236 meters per second and its linear acceleration is 0. The angular speed is approximately 26.18 radians per second and the angular acceleration is 0.

Key concepts

Angular Speed

Angular speed is a measure of how fast something spins around an axis. It is often expressed in radians per second. For circular motion, it's important to determine how quickly an object rotates. We use the angular speed to describe this.

• Angular speed (\( \omega \) in radians per second) tells us how many radians an object rotates through per second.
• To convert rotations per minute (rpm) to radians per second, multiply the rpm by the factor \( \frac{2\pi}{60} \). This is because one complete revolution is \( 2\pi \) radians and there are 60 seconds in a minute.

For instance, in the CD problem, the given speed is 250 rpm, which translates to an angular speed of approximately 26.18 radians per second. This allows us to see how fast the dirt spins along with the CD.

Linear Speed

Linear speed refers to how fast an object moves along a path. It's the rate at which the object's position changes over time.

• In rotational motion, the linear speed (\( v \)) can be calculated using the formula \( v = r\omega \), where \( r \) is the radius (distance from the center) and \( \omega \) is the angular speed.
• This relationship shows that the further from the center an object is, the faster its linear speed will be for the same angular speed.

In the case of the CD problem, the dirt particle is 2.0 cm from the CD's center. Converting this to meters gives 0.02 m. With an angular speed of 26.18 radians per second, its linear speed is approximately 0.5236 meters per second. This provides a sense of how quickly it moves along its circular path.

Angular Acceleration

Angular acceleration measures how quickly the angular speed of an object changes. It's given in radians per second squared.

• An object has angular acceleration if its rotational speed changes over time.
• In scenarios with constant rotational speed, like a CD player spinning a disk at a steady speed, the angular acceleration is zero.

In the exercise, since the CD rotates at a stable 250 rpm, the angular acceleration is 0. This means the angular speed does not change with time, indicating no need to overcome torque to increase or decrease speed.

Converting Units

Converting units is essential in physics to ensure calculations are consistent and understandable.

• Always convert lengths to meters, mass to kilograms, and time to seconds for calculations adhering to the International System of Units (SI).
• To convert rotational speed from rpm to radians per second, use \( 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/sec} \).
• For distances, converting cm to meters is done by dividing the centimeter value by 100, as there are 100 cm in a meter.

In our CD problem, we converted 2.0 cm to 0.02 m, and 250 rpm to 26.18 rad/s. Proper unit conversion avoids errors in problems and aids in understandable results.