how long is the groove on one side of a long play(33 1/3 rpm) phonograph record? assume there is a single recording and the author (beginning) groove is 5.75 inches from the center and the inner(ending) groove is 1.75 inches from the center. the recording plays for 23 minutes.
23 min * 100/3 rev/min = 766.666 rev
So, divide the distance between the 1st and last grooves into 767 grooves, and you get
4/766.666 = 0.0052 inches between grooves
So, you have an arithmetic sequence with
a = 5.75*2π
d = 0.0052*2π
and you want S767
Just plug and chug
You would have to use 1.75*2pi for a or -0.0052 for d
To determine the length of the groove on one side of a long play (33 1/3 rpm) phonograph record, we need to know the linear velocity of the groove and the time it takes to play the recording.
First, let's calculate the linear velocity of the groove. At 33 1/3 rpm (revolutions per minute), the record completes one full revolution every (1/33 1/3) minutes. This is equivalent to 1.8 seconds per revolution (60 seconds / 33.33... revolutions).
The linear velocity of the groove is calculated by multiplying the circumference of the groove by the number of revolutions it completes in a given time. The formula for circumference is C = 2πr, where C is the circumference and r is the radius.
In this case, the distance from the center to the author groove is 5.75 inches, and from the center to the inner groove is 1.75 inches. Since the groove is a spiral, its radius changes gradually from the center to the edge. To simplify the calculation, let's assume that the radius changes linearly.
So, the average radius (r_avg) of the groove can be calculated as (5.75 + 1.75) / 2 = 3.75 inches.
Now, let's convert the average radius to feet (since linear velocity is usually expressed in feet per second). Since there are 12 inches in a foot, the average radius in feet is 3.75 inches / 12 = 0.3125 feet.
To calculate the linear velocity, we need to multiply the circumference by the number of revolutions per second. Since there are 60 seconds in a minute, the number of revolutions per second is 1 / 1.8 revolutions (approximately 0.5556).
The linear velocity (v) can be calculated as v = C * (1 / T), where C is the circumference and T is the time in seconds. In this case, T is 23 minutes * 60 seconds = 1380 seconds.
Let's calculate the circumference:
C = 2πr_avg = 2 * 3.14159 * 0.3125 feet = 1.96 feet
Now, calculate the linear velocity:
v = 1.96 feet * 0.5556 (revolutions per second) = 1.089 feet per second
Finally, to determine the length of the groove, we multiply the linear velocity by the time:
Length = v * T = 1.089 feet/second * 1380 seconds = 1503.42 feet
Therefore, the groove on one side of the long play (33 1/3 rpm) phonograph record is approximately 1503.42 feet long when the recording plays for 23 minutes.