Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

A Blu-ray disc is approximately 12 centimeters in diameter. The drive motor of the Blu-ray player is able to rotate up to 10,000 revolutions per minute, depending on what track is being read. (a) Find the maximum angular speed (in radians per second) of a Blu-ray disc as it rotates. (b) Find the maximum linear speed (in meters per second) of a point on the outermost track as the disc rotates.

Step-by-Step Solution

Answer

The maximum angular speed of the Blu-ray disc is approximately 1047.197 radians per second, and the maximum linear speed of a point on the outermost track as the disc rotates is approximately 62.81 meters per second.

Step 1: Conversion of Revolutions per Minute to Radians per Second

Given the disc rotates at a speed of 10,000 revolutions per minute, we first need to convert this rotation speed into radians per second. One full rotation of the disc is equivalent to \(2\pi\) radians. Therefore, the angular speed in radians per minute is \(10,000 \times 2\pi\) radians per minute. There are 60 seconds in a minute, so to convert this to radians per second we divide by 60, giving us a result of \(\frac{10,000 \times 2\pi}{60}\) radians per second.

Step 2: Calculation of Angular Speed

Carrying out the above multiplication and division gives an angular speed of approximately 1047.197 radians per second.

Step 3: Calculation of Maximum Linear Speed

To find the maximum linear speed at the disc's outer edge, it is important to remember that linear speed equals radius times angular speed. We know the disc's diameter is 12 centimeters, which means the radius is 6 centimeters or 0.06 meters. Multiplying this radius by the angular speed gives a maximum linear speed of \(0.06 \times 1047.197\) meters per second.

Step 4: Final Result for Maximum Linear Speed

Performing the above multiplication gives us a maximum linear speed of approximately 62.81 meters per second.

Key Concepts

Angular Speed Conversion

Understanding angular speed conversion is essential in various fields of physics and engineering, where rotating systems are involved, such as in the case of a Blu-ray disc rotating in a player. Conversion is the process of translating a quantity expressed in one unit into another unit while maintaining the proportionate value. For example, converting revolutions per minute (rpm) of a disc into radians per second is important for mathematical and practical reasons, as radians are the standard SI unit of angular measure.

To convert rpm to radians per second, one must understand the relationship between these two units. A single revolution equates to one full circle, which is mathematically expressed as \(2\pi\) radians. Therefore, to convert from rpm to radians per second, you multiply by \(2\pi\) and then divide by 60 since there are 60 seconds in a minute. This process enables one to analyze the rotational motion in terms that can be applied across different disciplines and systems.

Practical applications, such as determining the correct speed profiles for industrial machinery or understanding the dynamics of rotating celestial bodies, all necessitate this form of conversion.

Radians Per Second Calculation

Radians per second is a rate that represents the angular velocity, indicating the change in the angular displacement with respect to time. When calculating radians per second, we are essentially determining how fast an object is rotating or spinning in terms of the angle through which it rotates every second.

A common scenario where this calculation is utilized is in the functioning of CD or Blu-ray players. For example, a Blu-ray disc rotating at 10,000 rpm can be calculated in radians per second by using the conversion factor previously discussed. Multiplying 10,000 rpm by \(2\pi\) will give the amount of radians the object covers in one minute. Dividing that number by 60 will then give the angular speed in radians per second. This calculation is vital for understanding the performance of the disc player and the stresses that components may experience during high-speed rotation.

Linear Speed Formula

The concept of linear speed becomes important when we need to relate the circular motion of an object to the speed at which a certain point on the object moves in a straight path. The linear speed formula connects an object's angular speed (\(\omega\)) with its radius (\(r\)) and is given by the equation \(v = \omega \times r\), where \(v\) is the linear speed.

In the context of a Blu-ray disc, applying this formula helps us understand how fast a point on the edge of the disc is moving in a linear fashion, even though the point is actually following a circular path. Given the radius of the disc and the angular speed in radians per second, one can calculate the maximum linear speed experienced by a point on the outer edge of the disc. This allows technicians and engineers to assess whether or not a particular component or material can withstand such speeds without being damaged or degraded over time.

Circular Motion in Physics

Circular motion refers to the movement of an object along the circumference of a circle at a constant distance from the center. It's a fundamental concept in physics that can describe the motion of satellites, planets, and many mechanical systems like wheels, gears, and as seen in our example, Blu-ray discs. There are two main types of quantities that describe circular motion: angular quantities (like angular speed and angular displacement) and linear quantities like linear speed and linear distance.

An understanding of circular motion includes knowing the relationships between these quantities. Every point on the rotating object has the same angular speed but the linear speed of a point depends on its distance from the center of rotation. Therefore, even though all points on a Blu-ray disc complete a full rotation together, points on the outer edge actually travel a greater distance and thus have a higher linear speed compared to points closer to the center. This insight is critical in designing and troubleshooting rotary systems to ensure they function as intended without causing undue stress or failure of the components involved.