Please answer questions 1-5 below. Use the equations on the side to answer quest
ID: 2040413 • Letter: P
Question
Please answer questions 1-5 below. Use the equations on the side to answer question 2
Question (write down the answers in Pre-lab (1) Regarding the transverse wave in Figure 1, what are the moving directions of points A, B, C, and D? (answer"+" for moving up, "-" for moving down, and "O" if not moving) (2) Apply the angle sum and difference identities on Eqs. (1) and (2) and derive Eq. (5), which is The up-and-down motion of the rope is perpendicular to the directior of the wave. Direction of wave (3) If a student varies a string's tension T and measures its wave velocity v, what regression Figure 1: Generating transverse waves on a strin analysis should be used to get the string's linear density u? What do the graph's slope and y-intercept mean and how can one read out u from them? (4) Regarding a string with constant tension T and linear density ?, please calculate the ratio of standing wave frequency between adjacent harmonic modes f2/f?, fs/f2, f4/f3 and fs/f4 (5) Regarding the standing wave in Figure 2, if you want to decrease the node number by 1 by Figure 2: Standing waves on a string changing only one of the four quantities-(i) oscillation frequency, (ii) string tension, (ii) string linear density, and (iv) string length-how should you do it and what is your reason? (Answer "increase" or "decrease" and explain your reason.) | |y. (x, t) = A sine-2nft), Eqs (1) y-(x, t)= Asin(T-x+2nft). Eqs. (2) 2? ysw(x, t)-y(x, t) + y_(x,t)-2A cos(2Tft) sinx. Eq. (5)Explanation / Answer
(1) Since the direction of up and down motion of the rope is perpendicular to the direction of wave, it is a transverse motion. Hence as per Figure 1, point A moves up ("+"), point B's motion gets cancelled hence it does not move ("0"), point C moves down ("-") and point D's motion also gets cancelled hence it does not move ("0").
(2) Lets consider y+(x,t) = A sin (P-Q)....(1)
y-(x,t) = A sin (P+Q)....(2)
Adding (1) and (2), we have,
y+(x,t) + y-(x,t) = A [sin (P-Q) + sin (P+Q)]
= A [(sin P * cos Q) - (cos P * sin Q) + (sin P * cos Q) + (cos P * sin Q)]
= A [2 sin P * cos Q] = 2A sin P * cos Q....(3)
Now, substituting P = (2? / ?)x
Q = (2? f t)
We derive from (3),
y+(x,t) + y-(x,t) = 2 A cos (2? f t) sin ((2? / ?)x)
(3) For a transverse wave motion, the wave velocity "v" is related to string's tension "T" and string's linear density "?" as,
v = ? (T / ?)....(3)
Or, v2 = T / ?
Or, ? = T / v2
When string tension is varied, the frequency "f" being the independent variable, is plotted along the abscissa while the wavelength "?" being the dependent variable, is plotted along the ordinate.
Thus from (3),
? = [? (T / ?)] * (1 / f)...4 ( since v = f * ? as per formula)
Thus, the slope of the graph is ? (T / ?) which is the wave velocity.
The y-intercept denotes the value of the wavelength "?" (say, ?0 ) when (1 /f ) is zero, hence infinite frequency and thus, at initial time.
Thus, ? can be found out from (4) using the straight line formula as,
? = [? (T / ?)] * (1 / f) + ?0
Or, T / ? = [ (? - ?0 ) * f ]2
Or, ? = T / [ (? - ?0 ) * f ]2
(4) Since standing waves on the string are sinusoidal, the number of waves are an integral multiple of half wavelength, i.e.,
L = n ? /2....(5) , where L = length of string,
n = number of node,
? = wavelength
Putting n = 1, 2, 3, 4, 5 in (5) and considering L is constant, we have,
? 1 / 2 = ? 2 , where ?1 = wavelength for n = 1 and ? 2 = wavelength for n =2
Or, f2 / f1 = ? 1 / ? 2 = 2 (since frequency "f" is inversely proportional to wavelength "?"), where f2 = frequency at n =2 and f1 = frequency at n =1
Similarly, 3 ? 3 / 2 = ? 2
Or, f3 / f2 = ? 2 / ? 3 = 3 / 2
Similarly, 4? 4 / 2 = 3 ? 3 / 2
Or, f4/ f3 = ? 3 / ? 4 = 4 / 3
Similarly, 5 ? 5 / 2 = 4 ? 4 / 2
Or, f5/ f4 = ? 4 / ? 5 = 5 / 4
(5) For decreasing number of nodes, the length of the string must be decreased as from (5), considering all parameters constant, the length of the string "L" is directly proportional to the number of nodes "n".