PLEASE SHOW ME STEP BY STEP (pretend like I\'m an idiot, because I sure feel lik
ID: 2152595 • Letter: P
Question
PLEASE SHOW ME STEP BY STEP (pretend like I'm an idiot, because I sure feel like one) AND EXPLAIN WHAT YOU ARE DOING. DO NOT COPY AND PASTE, BECAUSE I HAVE ALREADY READ EVERYTHING THAT IS ON HERE AND D0 NOT UNDERSTAND. I WILL RATE LIFESAVER FOR A GOOD AND COMPLETE EXPLANATION.
Ch16-50
For a certain transverse standing wave on a long string, an antinode is at x = 0 and an adjacent node is at x = 0.30 m. The displacement y(t) of the string particle at x = 0 is shown in Fig. 16-39, where the scale of the y axis is set by ys = 4.0 cm. When t = 0.90 s, what is the displacement of the string particle at (a)x = 0.50 m and (b)x = 0.50 m ? What is the transverse velocity of the string particle at x = 0.50 m at (c)t = 0.90 s and (d)t = 1.4 s?
Explanation / Answer
Harmonic waves travelling in opposite directions can be represented by the equations below:
y = y0 sin(kx - t)
and
y = y0 sin(kx + t)
So the resultant wave y equation will be the sum of y1 and y2:
y = y0 sin(kx - t) + y0 sin(kx + t)
Using the trigonometric sum-to-product identity for 'sin(u) + sin(v)' to simplify:
y = 2 y0 cos(t) sin(kx)
for this question,
T = 2* (1.2s) = 2.4s
w = 2/T = 2*3.1416/2.4 = 2.618 rad/s
= 2 * 0.30 m = 0.60 m
k = 2/ = 2*3.1416/0.60 = 10.472 m-1
a)
t = 0.90 s and x = 0.50 m
y = 2 y0 cos(t) sin(kx)
y = 2 * 0.04 * cos(2.618*0.90) sin(10.472*0.50)
y = 2 * 0.04 * (-0.70711) * (-0.866019)
y = 0.0490 m
b)
is x = 0.50 m again??!!
t = 0.90 s and x = 0.50 m
y = 2 y0 cos(t) sin(kx)
y = 2 * 0.04 * cos(2.618*0.90) sin(10.472*0.50)
y = 2 * 0.04 * (-0.70711) * (-0.866019)
y = 0.0490 m
c)
y = 2 y0 cos(t) sin(kx)
dy/dt = 2 y0 sin(t) sin(kx)
dy/dt = 2 * 0.04 * 2.618 * sin(2.618*0.90) sin(10.472*0.50)
dy/dt = 2 * 0.04 * 2.618 * 0.707103 * (-0.866019)
dy/dt = -0.128 m/s
d)
y = 2 y0 cos(t) sin(kx)
dy/dt = 2 y0 sin(t) sin(kx)
dy/dt = 2 * 0.04 * 2.618 * sin(2.618*1.4) sin(10.472*0.50)
dy/dt = 2 * 0.04 * 2.618 * (-0.5)* (-0.866019)
dy/dt = -0.0907 m/s