Problem 4 - College Physics

Question

The once-popular LP (long-play) records were 12 in. in diameter and turned at a constant \(33 \frac{1}{3} \mathrm{rpm} .\) Find (a) the angular speed of the LP in rad \(/ \mathrm{s}\) and \((\mathrm{b})\) its period in seconds.

Step-by-Step Solution

Answer

(a) \( \frac{10\pi}{9} \) rad/s, (b) 1.8 seconds.

Step 1: Convert Revolutions to Radians

To find the angular speed in radians per second, we first need to convert revolutions per minute (rpm) to radians per minute. Since one revolution is equal to \( 2\pi \) radians, we multiply 33 1/3 rpm by \( 2\pi \) to get the angular distance in radians per minute: \( 33\frac{1}{3} \times 2\pi = \frac{100}{3} \times 2\pi = \frac{200\pi}{3} \) radians per minute.

Step 2: Convert Minutes to Seconds

Now, we need to convert the angular speed from radians per minute to radians per second. There are 60 seconds in one minute, so we divide the angular speed by 60: \( \frac{200\pi}{3} \div 60 = \frac{200\pi}{180} = \frac{10\pi}{9} \) radians per second. This is the angular speed we are looking for in part (a).

Step 3: Find the Period

The period is the time it takes to complete one full revolution. We start by finding the number of revolutions per second by dividing the revolutions per minute by 60: \( 33\frac{1}{3} \div 60 = \frac{1}{1.8} = \frac{3}{5.4} = \frac{3}{3.2} \approx 0.5556 \) revolutions per second. The period \( T \) is the reciprocal of this frequency, so \( T \approx \frac{1}{0.5556} \approx 1.8 \) seconds.

Key Concepts

LP Records

LP records, known as long-play records, revolutionized the music industry in the mid-20th century. These records are 12 inches in diameter and were a popular medium for music listening. One unique feature was their rotational speed, spinning at 33 1/3 revolutions per minute (RPM). LP records could hold more music due to the way they were manufactured. This allowed artists to feature entire albums on a single disc. It became a cultural icon, influencing music production and distribution before the digital era began. Understanding the mechanics of LPs involves engaging with concepts like angular speed, as we often calculate how fast the records spin when played.

Radians Conversion

To discuss angular speed, we need to understand radians. A radian is a unit of angular measure used in mathematics. In terms of circular motion, one complete revolution is equal to the angle produced by the radius creating the circumference of a circle.A full circle in radians is equal to \(2\pi\) radians. This conversion is crucial when dealing with revolving bodies, like LP records. We often need to transform revolutions into radians, as radians offer a practical unit for describing angular velocity.Radians make calculations involving angles and rotation straightforward, since \(2\pi\) simplifies the relationship between a circle’s radius and its circumference.

Revolutions Per Minute (RPM)

Revolutions per minute (RPM) is a common unit of rotational speed. It tells us how many complete turns an object makes in one minute. For LP records, a standard speed of 33 1/3 RPM meant the record turned precisely 33.33 times in one minute. To calculate angular speed in radians per second, understanding RPM is essential. It's the starting point for conversions and calculations needed to find an LP record’s angular speed in different units. Using RPM, one can determine aspects like how fast a record spins or how long it takes to complete a revolution, making it helpful for solving problems in physics and engineering related to rotational dynamics.

Revolutions to Radians Conversion

Converting revolutions to radians is a key step in determining angular speed. Since one revolution is equivalent to \(2\pi\) radians, to find the angular speed in radians per minute for an LP spinning at 33 1/3 RPM, we multiply \(33\frac{1}{3}\) by \(2\pi\).Subsequently, converting this value from radians per minute to radians per second involves dividing by 60, because there are 60 seconds in a minute. In our exercise, this results in an angular speed of \(\frac{10\pi}{9}\) radians per second.This conversion is not just essential for understanding LP records but also for analyzing any object that rotates, making radians a fundamental aspect of rotational motion studies.