Standing waves on a string are generated by oscillations having amplitude 0.005 m, angular frequency 942 rad/s, and wave number 0.750p rad/m.
a.) What is the equation of the standing wave?
b.) At what distances from x=0 are the nodes and antinodes?
c.) What is the frequency of a point on the string at an antinode?
d.) If the string is 4m long, how many nodes are there?
y(x,t) =|2•A•coskx|•cosωt,
y(x,t) =|2•0.005•cos(0.750•π•x)| •cos(942•t),
Antinods:
x=m• (λ/2), m=0,1,2,...
The 1st antinode at the origin (m=0),
the 2nd antinode at λ/2 (m=1).
Distance from origin = λ/2
Nodes:
x= (m+½)• (λ/2).
The 1st node at λ/4 (m=0),
the 2nd antinode at 3λ/4 (m=1)
Distance from origin = λ
The frequency is f=ω/2π=942/2 π =149.9 Hz.
λ=2π/k=2π/0.75 π=8/3 (meters).
If the antinode is at the origin, then we have
3 nodes on the distance 4 m at the points:
x=λ/4, 3λ/4, 5λ/4,
and 4 antinodes at
x= 0, λ/2, λ, 3λ/2.
To answer these questions, we need to understand the properties of standing waves and how to determine their equation, nodes, antinodes, and frequency.
a.) The equation of a standing wave on a string can be expressed as:
y(x, t) = A*sin(kx)*cos(ωt)
where:
- y is the displacement of the string from its resting position,
- x is the position along the string,
- t is the time,
- A is the amplitude of oscillation,
- k is the wave number (2π/λ),
- ω is the angular frequency (2πf), and
- f is the frequency.
Given the values, A = 0.005 m, ω = 942 rad/s, and k = 0.750π rad/m, we can substitute these values into the equation to get:
y(x, t) = 0.005*sin(0.750π*x)*cos(942t)
b.) In a standing wave, nodes are points where the displacement is always zero (minimum points), and antinodes are points where the displacement is maximum. To determine the positions of nodes and antinodes (distances from x = 0), we can use the formula:
x_node = (n * λ) / 2
x_antinode = (2n + 1) * (λ / 4)
where n is an integer representing the harmonic number and λ is the wavelength (2π/k).
Given k = 0.750π rad/m, we can calculate the wavelength:
λ = 2π / k
= 2π / (0.750π)
= 8/3 m
Now, we can use this value to find the positions of nodes and antinodes.
For nodes:
x_node = (n * λ) / 2
= (n * (8/3)) / 2
= (4/3) * n
For antinodes:
x_antinode = (2n + 1) * (λ / 4)
= (2n + 1) * ((8/3) / 4)
= (2n + 1) * (2/3)
c.) The frequency of a point on the string at an antinode is the same as the frequency of the oscillation, which is given as ω = 942 rad/s. The frequency can be calculated as:
f = ω / (2π)
= 942 / (2π)
≈ 149.8 Hz
d.) If the string is 4m long, a node occurs at each end of the string, so we need to consider the number of nodes in the interior. The distance between adjacent nodes is half a wavelength. For a string of length L, the number of nodes is given by:
Number of nodes = L / (λ / 2)
Given L = 4 m and λ = 8/3 m, we have:
Number of nodes = (4) / ((8/3) / 2)
= 3
Therefore, there are 3 nodes in the interior of the string.
Overall, the answers to the questions are:
a.) The equation of the standing wave is y(x, t) = 0.005*sin(0.750π*x)*cos(942t).
b.) The positions of the nodes are (4/3) * n from x = 0, and the positions of the antinodes are (2n + 1) * (2/3) from x = 0.
c.) The frequency at an antinode is approximately 149.8 Hz.
d.) There are 3 nodes in the interior of the 4m long string.