A sinusoidal wave in a rope is described by
the wave function
y = A sin(k x + ! t) ,
where A = 0.47 m, k = 1 m−1, ! = 26 Hz, x
and y are in meters, and t is in seconds.
vibrator
26 Hz
ì = 3.8 g/m
m
18.8496 m
If the tension in the rope is provided by
an arrangement like the one illustrated above,
what is the value of the suspended mass? The acceleration of gravity is 9.8 m/s2 . The rope
has a linear mass density of 3.8 g/m.
How can we answer this without some description of the rope mechanism.
rope has length of 18.8496 with the oscilator on one end and the mass end hanging
To find the value of the suspended mass, we can use the equation for the velocity of a wave on a rope:
v = sqrt(T/µ)
Where:
v is the velocity of the wave (m/s),
T is the tension in the rope (N),
and µ is the linear mass density of the rope (kg/m).
Since we are given the frequency of the wave (f) and the wave number (k), we can find the velocity using the formula:
v = f / k
Let's find the velocity first.
To find the value of the suspended mass, we need to consider the forces acting on the rope and determine the relationship between tension, linear mass density, and the wave.
We know that the tension in the rope is responsible for providing the restoring force that allows the wave to propagate. The wave function describes the displacement of the rope at any point (x) and time (t).
The restoring force is related to the tension (T) and the linear mass density (µ) of the rope. The linear mass density is the mass of the rope divided by its length. In this case, we are given that the linear mass density is 3.8 g/m.
The wave function also includes the wavenumber (k) and angular frequency (!). The wavenumber is related to the wavelength (λ) of the wave and is given by k = 2π/λ, where λ is the distance between two consecutive peaks or troughs of the wave.
The angular frequency (!) is related to the frequency (f) of the wave by ! = 2πf.
Now, let's determine the relationship between tension, linear mass density, and the wave:
The wave speed (v) is given by v = !/k. In this case, v = 26 Hz / 1 m^(-1) = 26 m/s.
The wave speed is also related to the tension and linear mass density by v = √(T/µ).
Equating the two expressions for the wave speed, we have:
√(T/µ) = 26 m/s.
Squaring both sides, we get:
T/µ = (26 m/s)^2.
We can now find the tension (T) in the rope.
T = µ * (26 m/s)^2.
The tension in the rope is responsible for supporting the weight of the suspended mass. The weight (W) is given by W = m * g, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2).
Since T = W, we have:
m * g = µ * (26 m/s)^2.
Rearranging the equation, we can solve for the mass (m):
m = µ * (26 m/s)^2 / g.
Plugging in the given linear mass density (µ = 3.8 g/m) and acceleration due to gravity (g = 9.8 m/s^2), we get:
m = (3.8 g/m) * (26 m/s)^2 / (9.8 m/s^2).
m = 805.9 g / 9.8 kg/m^2.
Converting grams to kilograms:
m = 0.8059 kg.
Therefore, the value of the suspended mass is approximately 0.8059 kg.