Answer:
The resonant frequency of a column of air in a resonance tube is given by:
f = (n * v) / (4 * L)
where n is the harmonic number, v is the velocity of sound in air, and L is the length of the tube.
To find the next resonating length, we need to determine the harmonic number for the next mode of vibration. Since the current length resonates at the fundamental frequency (n=1), the next resonant length will correspond to the second harmonic (n=2).
Using the formula above, we can solve for the next resonant length:
f = (2 * v) / (4 * L')
where L' is the next resonant length.
We can rearrange this formula to solve for L':
L' = (2 * v) / (4 * f)
We are given that the end correction is 1.0 cm, so we need to subtract this from the resonant lengths we calculate.
Plugging in the values we know:
v = 343 m/s (the speed of sound in air at room temperature)
f = the frequency of the tuning fork (not given)
L = 15.0 cm - 1.0 cm = 14.0 cm
L' = (2 * 343 m/s) / (4 * f)
L' = 0.171 m / f
Now, we can substitute this expression for L' back into the equation for the next resonant length:
L2' = 0.171 m / f * 2
L2' = 0.342 m / f
Finally, we can convert this to centimeters and add back the end correction:
L2' = 34.2 cm / f + 1.0 cm
So, the next resonating length will be 34.2 cm divided by the frequency of the tuning fork, plus 1.0 cm.