A 3.49 rad/s (33 rpm) record has a 5.99-kHz tone cut in the groove. If the groove is located 0.100 m from the center of the record, what is the "wavelength" in the groove?
To find the wavelength in the groove of a rotating record, we need to understand the relationship between the rotational speed of the record and the frequency of the tone cut in the groove.
First, let's convert the rotational speed from rpm (revolutions per minute) to rad/s (radians per second). We can use the conversion factor: 1 rpm = π/30 rad/s.
Given:
Rotational speed = 3.49 rad/s (33 rpm)
Groove distance from center = 0.100 m
Frequency of the tone = 5.99 kHz
Step 1: Convert rpm to rad/s
3.49 rad/s = (33 rpm) * (π/30 rad/s) = 3.63 rad/s
Step 2: Calculate the circumference of the groove
The circumference of a circle is given by the formula:
C = 2πr
where r is the radius of the circle (groove distance from the center).
C = 2π * 0.100 m
Step 3: Calculate the distance traveled by the groove in one revolution
The distance traveled by the groove in one revolution is equal to the circumference of the groove.
Distance in one revolution = C = 2π * 0.100 m
Step 4: Calculate the wavelength in the groove
Wavelength represents the distance covered by one complete cycle of the tone in the groove, which is equal to the distance traveled by the groove in one revolution.
Wavelength = Distance in one revolution
Now, let's calculate the values:
C = 2π * 0.100 m
≈ 0.628 m
Wavelength ≈ 0.628 m
Therefore, the wavelength in the groove of the record is approximately 0.628 meters.