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A student investigated the variation of the fundamental frequency $f$ of a stretched string with its tension $T$ - Leaving Cert Physics - Question 3 - 2009
Question 3
A student investigated the variation of the fundamental frequency $f$ of a stretched string with its tension $T$. The following is an extract of the student’s accoun... show full transcript
Worked Solution & Example Answer:A student investigated the variation of the fundamental frequency $f$ of a stretched string with its tension $T$ - Leaving Cert Physics - Question 3 - 2009
Step 1
How was the tension measured? How did the student know that resonance occurred?
Answer
The tension in the string was measured using a newton balance or scales, which allowed for accurate readings of the weight of the pan and its contents. The student knew that resonance had occurred when the string vibrated at maximum amplitude, which was evident as the loudest sound produced. This dramatic increase in amplitude indicates that the vibrating tuning fork matched the fundamental frequency of the string, leading to resonance.
Step 2
Draw a suitable graph to show the relationship between the fundamental frequency of a stretched string and its tension.
Answer
To illustrate the relationship, plot the provided data points (frequency) against (tension) on a graph. Ensure that:
- Both axes are correctly labeled with frequency on the y-axis and tension on the x-axis.
- Points plotted accurately reflect the recorded data.
- Draw a straight line that best fits the data points, which should ideally be a linear relationship.
The relationship is that frequency is proportional to the square root of tension , expressed mathematically as:
f ext{ is proportional to } rac{1}{ ext{s}} imes rac{1}{ ext{L}} imes ext{T}
This linear graph will go through the origin as tension increases, which confirms the relationship.
Step 3
Use your graph to (i) estimate the fundamental frequency of the string when its tension is 11 N;
Answer
From the graph, estimate the corresponding frequency when tension is 11 N. Using the trend, it can be inferred that the frequency is approximately Hz. A more accurate estimate can be calculated by extrapolating from the straight line equation derived from the plotted points.
Step 4
(ii) calculate the mass per unit length of the string.
Answer
To find the mass per unit length of the string, use the relationship between frequency , tension , and length of the string:
ho} $$ Given that for fundamental frequency, frequency can be plugged into this equation, rearranging gives:ho = rac{T}{(2Lf)^2} $$ Substituting appropriate values into this formula (length m for 40 cm), leads to:
ho = 5.86 imes 10^{-6} ext{ kg m}^{-1} $$