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.
Acceleration due 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.
The speed of the transverse wave along the string is 340 m/s.
The amount of suspended mass needed to double the number of loops in the standing wave pattern is 0.072 kg.
The amplitude of the standing wave is 4.5 cm.
To find the speed of the transverse wave along the string, we can use the formula:
v = √(T/μ)
where v is the wave speed, T is the tension in the string, and μ is the linear density of the string.
To determine the tension in the string, we can use the relation between the mass and the tension in a vertical hanging string:
T = Mg
where M is the mass attached to the string and g is the acceleration due to gravity.
In this case, the mass (M) is not given, but we know the number of loops (8). Each loop corresponds to an antinode. The distance between consecutive antinodes is λ/2, where λ is the wavelength of the standing wave. The length from the tuning fork to the top of the pulley is given as 1.1 m.
Since the standing wave pattern has 8 loops, there will be 16 antinodes. Given that λ/2 is the distance between consecutive antinodes and the total length of the string is 1.1 m, we can calculate the wavelength of the wave:
λ = 2 × (1.1 m) / 16
Now, using the formula for wave speed:
v = λf
where f is the frequency of the tuning fork, which is given as 120 Hz. Substitute the calculated wavelength into this equation to find the speed of the transverse wave along the string.
For the second part of the question, to double the number of loops from 8 to 16, we need to determine the new mass (M). Remember that the tension in the string increases with the mass. Let's denote the original mass as M1 and the new mass as M2. Since the wave speed is directly proportional to the square root of the tension, we can write:
v1/v2 = √(T1/T2)
Simplifying the equation by squaring both sides:
(v1/v2)^2 = T1/T2
Since we are doubling the number of loops (antinodes), we can set up the equation:
(8/16)^2 = T1/T2
Solve this equation to find the ratio of T1 to T2. Then, use the relation T = Mg to calculate the new mass M2.
For the third part of the question, we need to find the amplitude of the standing wave. The amplitude is the maximum displacement from the rest position. In a standing wave, the amplitude is related to the distance traveled by a point on the string during one complete cycle.
The distance traveled by a point on the string during one complete cycle is equal to the wavelength (λ). We already calculated the wavelength in the first part of the question. So, the amplitude of the standing wave would be half of the wavelength.
Amplitude = λ/2
Substitute the calculated wavelength into this equation to find the amplitude in centimeters.