A transverse wave is formed on a long skipping rope. The equation of the transverse wave is y=5.50 sin(0.031π x+5.52π t) where x and y are in centimeters, t is in seconds, and Π is pi (3.14..). Find the amplitude of the wave? what is the wavelength of the transverse wave? Find the frequency of the wave?calculate the speed of the wave?calculate the transverse displacement when x=3.9cm and t=0.29s?
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http://www.insula.com.au/physics/1279/L14.html
To find the amplitude of the wave, we look at the coefficient in front of the sine function. In this case, the amplitude is 5.50 cm.
The equation for a transverse wave can be written as y = A sin(kx - ωt), where A is the amplitude, k is the wave number (2π/λ), ω is the angular frequency (2πf), and λ is the wavelength.
From the given equation, we can see that the wave number is 0.031π and the angular frequency is 5.52π.
To find the wavelength, we can use the formula λ = 2π/k. Plugging in the wave number, we get λ = 2π/(0.031π) = 200 cm.
To find the frequency, we use the formula f = ω/(2π). Plugging in the angular frequency, we get f = 5.52π/(2π) = 2.76 Hz.
The speed of the wave can be calculated using the formula v = λf. Plugging in the wavelength and frequency, we get v = 200 cm * 2.76 Hz = 552 cm/s.
To calculate the transverse displacement when x = 3.9 cm and t = 0.29 s, we substitute these values into the equation. Plugging in x = 3.9 cm and t = 0.29 s, we get y = 5.50 sin(0.031π * 3.9 + 5.52π * 0.29) = 5.50 sin(0.12π + 1.60π) = 5.50 sin(1.72π).
Note: π = 3.14 approximately.
To calculate the value of sin(1.72π), we convert it to degrees by multiplying by 180/π. sin(1.72 * 180/π) ≈ sin(98.183°) ≈ 0.9985. Therefore, the transverse displacement at x = 3.9 cm and t = 0.29 s is approximately 5.50 * 0.9985 = 5.492 cm.
Therefore, the answers are:
Amplitude: 5.50 cm
Wavelength: 200 cm
Frequency: 2.76 Hz
Speed: 552 cm/s
Transverse displacement at x = 3.9 cm and t = 0.29 s: 5.492 cm
To find the amplitude of the wave, you can refer to the equation you were given: y = 5.50 sin(0.031π x + 5.52π t). In this equation, the amplitude is the coefficient of the sine function, which is 5.50. Therefore, the amplitude of the wave is 5.50 centimeters.
To find the wavelength of the transverse wave, you need the coefficient in front of 'x' in the equation. In this case, it is 0.031π. The wavelength (λ) is related to this coefficient by the formula λ = 2π/k, where k is the coefficient. So, in this case, the wavelength is given by λ = 2π/(0.031π) = 2/0.031 = 64.52 centimeters.
Next, to find the frequency of the wave, you can use the equation v = fλ, where v is the wave velocity, f is the frequency, and λ is the wavelength. In the given equation, the velocity of the wave is not explicitly mentioned, so we can assume it to be the speed of light (3 x 10^8 meters per second). Therefore, we have 3 x 10^8 = f * 0.6452 (converting cm to meters). Solving for f, we get f = (3 x 10^8) / 0.6452 = 464,639.40 Hz.
To calculate the speed of the wave, you can use the formula v = fλ, where v is the wave velocity, f is the frequency, and λ is the wavelength. Plugging in the values we have, v = (464,639.40 Hz) * (0.6452 m) = 299,864.77 meters per second.
Finally, to calculate the transverse displacement when x = 3.9 cm and t = 0.29 seconds, you can substitute those values into the given equation. Plugging in the values, we have y = 5.50 * sin[(0.031π * 3.9) + (5.52π * 0.29)]. Simplifying this expression, we get y = 5.50 * sin(0.121π + 1.6068π). Calculate the value within the sine function, y = 5.50 * sin(1.727π). Use a calculator to evaluate this, and you will find the transverse displacement at that point in the wave.