To demonstrate standing waves, one end of a string is attached to a tuning fork with frequency 120 Hz. The other end of the string passes over a pulley and is connected to a suspended mass M, as shown.
The value of M is such that the standing wave pattern has eight loops “anti-nodes”. The length of the string from the tuning fork to the point where the string touches the top of the pulley is 1.1 m. The linear density of the string is 8×10−5 kg/m, and remains constant throughout the experiment.
The acceleration to gravity is 9.8m/s^2
Determine the speed of the transverse wave along the string. Answer in units of m/s.
The speed of waves along the string in- creases with increasing tension in the string.
In order to double the number of loops in the standing wave pattern (so that there are 16 loops in total), how much suspended mass should we use? Answer in units of kg.
If a point on the string at an antinode travels a total distance of 9 cm during one complete cycle, what is the amplitude of the standing wave?
Answer in units of cm.
2.25
To determine the speed of the transverse wave along the string, we can use the formula:
v = √(T/μ)
Where:
- v is the speed of the wave
- T is the tension in the string
- μ is the linear density of the string
The frequency of the tuning fork is given as 120 Hz, so the period of the wave is 1/120 s. The number of loops, or antinodes, in the standing wave pattern is 8. The distance from the tuning fork to the pulley is given as 1.1 m.
To find the tension in the string, we can use the following formula:
T = (Mg)/n
Where:
- T is the tension in the string
- M is the mass suspended from the string
- g is the acceleration due to gravity
- n is the number of loops in the standing wave pattern
Substituting the given values, we can solve for T:
T = (Mg)/8
We can then substitute T and μ into the equation for v to find the speed of the wave:
v = √(((Mg)/8)/μ)
To double the number of loops, we need to find the new mass, M'. Let M' be the new mass needed.
Using the equation for tension, we can say:
T' = (M'g)/16
To determine the new mass, M', we can equate T and T':
(Mg)/8 = (M'g)/16
Simplifying and solving for M':
M' = (M/2)
In other words, we need half the mass to double the number of loops in the standing wave pattern.
Finally, to calculate the amplitude of the standing wave, we can use the formula:
Amplitude = (Total distance traveled)/(2n)
Where:
- Amplitude is the amplitude of the standing wave
- Total distance traveled is given as 9 cm
- n is the number of loops in the standing wave pattern, given as 8
Substituting the values, we can find the amplitude:
Amplitude = 9 cm / (2 * 8) = 9/16 cm
Now we have all the answers:
1. The speed of the transverse wave along the string is calculated as v = √(((Mg)/8)/μ), where the values of M, g, and μ are given. Calculate this to find the speed of the wave in m/s.
2. To double the number of loops, we need to use half the mass M. Calculate this value to find the new suspended mass M' in kg.
3. The amplitude of the standing wave is calculated as Amplitude = (Total distance traveled) / (2n), where the values of the total distance traveled and n are given. Calculate this to find the amplitude in cm.
To determine the speed of the transverse wave along the string, we can use the formula:
v = √(T/μ)
where v is the speed of the wave, T is the tension in the string, and μ is the linear density of the string.
First, let's find the tension in the string. The tension in the string is equal to the weight of the suspended mass. The weight of an object is given by:
F = m * g
where F is the weight, m is the mass, and g is the acceleration due to gravity.
Since the mass M is suspended from the string, the tension in the string is equal to the weight of M:
T = M * g
Given that g = 9.8 m/s^2 and M is yet to be determined, we'll solve for T later.
Next, let's find the linear density of the string. The linear density is given as 8×10^(-5) kg/m.
Now, substituting the known values into the formula for the speed of the wave:
v = √(T/μ)
we have:
v = √((M * g) / μ)
To double the number of loops in the standing wave pattern, we need to determine the mass M. The number of loops is proportional to the frequency of the wave and the length of the string. We can use the equation:
n = 2Lw / λ
where n is the number of loops, L is the length of the string, w is the frequency of the wave, and λ is the wavelength.
Given that the frequency of the tuning fork is 120 Hz and the length of the string is 1.1 m, and we want to double the number of loops, we can rearrange the equation as:
n / 2 = Lw / λ
Substituting the known values:
8 / 2 = 1.1 * 120 / λ
Simplifying:
4 = 132 / λ
Cross-multiplying:
4λ = 132
Solving for λ:
λ = 132 / 4 = 33 m
Now that we know the wavelength, we can calculate the speed of the wave. The speed of the wave is given by the formula:
v = λ * w
Substituting the values:
v = 33 * 120 = 3960 m/s
Now, using the formula for the speed of the wave:
v = √((M * g) / μ)
we can solve for M:
(M * g) / μ = v^2
Substituting the known values:
(M * 9.8) / (8×10^(-5)) = (3960)^2
Cross-multiplying:
M * 9.8 = (3960)^2 * (8×10^(-5))
Simplifying:
M = (3960)^2 * (8×10^(-5)) / 9.8
Calculating:
M ≈ 128.49 kg
To find the amplitude of the standing wave, we need to know the distance traveled by a point on the string at an antinode during one complete cycle. The distance traveled is equal to half the wavelength of the wave:
distance = λ/2
Given that the distance is 9 cm, which is equal to 0.09 m, we can set up the equation:
λ/2 = 0.09
Simplifying:
λ = 0.18
The amplitude of the standing wave is equal to half the wavelength:
amplitude = λ / 2 = 0.18 / 2 = 0.09 m
Converting the amplitude to cm:
amplitude = 0.09 m * 100 cm/m = 9 cm
Therefore, the amplitude of the standing wave is 9 cm.