This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Hey, good to see you again. Here is the solution for the experiment "Measuring the Speed of Sound in Trapped Air": Diagram [scale=0.8] % Beaker (0,0) -- (0,5) -- (4,5) -- (4,0) -- cycle; [fill=blue!20] (0.1,0.5) rectangle (3.9,2.5); % Water level at (2,1.5) Water; % Pipe [thick] (1.5,2.5) -- (1.5,6.5); [thick] (2.5,2.5) -- (2.5,6.5); [thick] (1.5,6.5) arc (180:0:0.5); % Top arc [thick] (1.5,2.5) arc (180:360:0.5); % Bottom arc (hidden) at (2,4.5) Pipe; % Air column [dashed] (2.5,2.5) -- (2.5,6.5); % Inner line for air column [<->] (2.8,2.5) -- (2.8,6.5) node[right, midway] L; at (2,5.5) Air Column; % Tuning Fork [thick] (5,6) -- (5,6.5); [thick] (5,6.5) -- (5.5,6.5); [thick] (5,6.5) -- (4.5,6.5); [thick] (5,6) -- (5,5.5); [thick] (5,5.5) arc (90:270:0.25); at (5,7) Tuning Fork; % Sound waves [decorate,decoration=snake,amplitude=.5mm,segment length=2mm] (4.5,6.5) -- (2,6.5); [decorate,decoration=snake,amplitude=.5mm,segment length=2mm] (5.5,6.5) -- (2,6.5); Procedure 1. Set up the apparatus by placing the plastic pipe vertically into a tall measuring cylinder partially filled with water. 2. Measure the room temperature using a thermometer and the internal diameter of the pipe using a vernier caliper (or use the given value). 3. Strike a tuning fork of known frequency (e.g., 512 Hz) gently on a rubber pad and hold it horizontally just above the open end of the pipe. 4. Adjust the water level in the pipe by slowly raising or lowering the pipe in the cylinder until a loud resonant sound is heard. This corresponds to the first resonance. 5. Measure the length of the air column (L_1) from the top of the pipe to the water surface using a metre rule. Record this value. 6. Continue to lower the water level until a second, louder resonance is heard. 7. Measure the length of this air column (L_2) from the top of the pipe to the water surface. Record this value. 8. Repeat steps 4-7 several times to obtain multiple readings for L_1 and L_2, and calculate their average values. Observations/Results Frequency of tuning fork, f = 512 Hz Internal diameter of the pipe, d = 2.5 cm = 0.025 m Room temperature, T = 25^ C | Reading | First Resonance Length (L_1) | Second Resonance Length (L_2) | | :------ | :----------------------------- | :------------------------------ | | 1 | 16.1 cm | 49.9 cm | | 2 | 16.2 cm | 49.8 cm | | 3 | 16.0 cm | 50.0 cm | | Average | 16.1 cm | 49.9 cm | Average L_1 = 16.1 cm = 0.161 m Average L_2 = 49.9 cm = 0.499 m Calculations Step 1: Calculate the wavelength () of the sound wave. The difference between consecutive resonance lengths corresponds to half a wavelength: = 2(L_2 - L_1) Substitute the average values: = 2(0.499 m - 0.161 m) = 2(0.338 m) = 0.676 m Step 2: Calculate the speed of sound (v) using the formula v = f. v = 512 Hz × 0.676 m v = 346.112 m/s Step 3: Calculate the theoretical speed of sound at the measured temperature for comparison. The speed of sound in air at temperature T (in ^ C) is approximately given by: v_T = 331 m/s + (0.6 m/s/^ C × T) For T = 25^ C: v_25 = 331 m/s + (0.6 m/s/^ C × 25^ C) v_25 = 331 m/s + 15 m/s v_25 = 346 m/s Inference The speed of sound in air at 25^ C was determined to be approximately 346.1 m/s. Conclusion The experiment successfully determined the speed of sound in air at room temperature (25^ C) to be 346.1 m/s. This value is in close agreement with the theoretical value of 346 m/s for sound in air at 25^ C. Precautions Ensure the tuning fork is struck gently on a rubber pad to produce a clear, pure tone and avoid damage. Hold the vibrating tuning fork horizontally and as close as possible to the open end of the pipe to maximize the transfer of sound energy. Ensure the pipe is held perfectly vertical to maintain a consistent and accurate length of the air column. Take multiple readings for the resonance lengths (L_1 and L_2) and calculate their averages to minimize random errors. Accurately identify the point of maximum loudness for resonance, as this indicates the precise resonance length. Read the metre rule at eye level to avoid parallax errors when measuring the air column lengths. Record the room temperature accurately using a thermometer, as the speed of sound is temperature-dependent.