Angular velocity, \(\omega = 33.3\) RPM = \(33.3 \times \frac{2\pi}{60} = 3.49\) rad/s
Time to play one side, \(t = 25\) min = \(25 \times 60 = 1500\) s
To find:
Number of grooves on each side, \(n\)
The linear velocity of the record at the outermost groove is given by:
$$v = \omega R$$
Where \(R\) is the radius of the record.
The circumference of the record at the outermost groove is:
$$C = 2\pi R$$
The number of grooves on each side is equal to the circumference of the record divided by the groove spacing:
$$n = \frac{C}{d}$$
Where \(d\) is the groove spacing.
Substituting the expressions for \(C\) and \(v\) into the equation for \(n\), we get:
$$n = \frac{2\pi R}{\omega t}$$
Substituting the given values, we get:
$$n = \frac{2\pi \times 0.15 \ m}{3.49 rad/s \times 1500 s}$$
$$n \approx 1100 \text{ grooves}$$
Therefore, each side of the LP record has approximately 1100 grooves.