Transverse waves with a speed of 50.0 m/s are to be produced on a stretched string. A 5.00m length of string with a total of 60 grams is used.
a) what is the required tension in the string?
b) calculate the wave speed in the string if the tension is 8.00n.
velocity= sqrt (tension/(mass/length))
Tension=velocity^2 *mass/length
Let's solve this step by step:
a) To find the required tension in the string, we can use the wave speed formula:
Wave speed (v) = √(tension (T) / linear mass density (μ))
The linear mass density of the string (μ) is calculated by dividing the total mass of the string by its length:
μ = mass / length
Given that the length of the string is 5.00m and the total mass is 60 grams, we convert the mass to kilograms by dividing it by 1000:
mass = 60 grams = 60/1000 kg = 0.06 kg
Now we can calculate the linear mass density:
μ = 0.06 kg / 5.00 m = 0.012 kg/m
Next, we can substitute the values into the wave speed formula to find the tension:
50.0 m/s = √(T / 0.012 kg/m)
Squaring both sides of the equation:
(50.0 m/s)^2 = (T / 0.012 kg/m)
2500 m^2/s^2 = T / 0.012 kg/m
Multiply both sides by 0.012 kg/m:
T = 2500 m^2/s^2 * 0.012 kg/m = 30 N
The required tension in the string is 30 N.
b) To calculate the wave speed in the string when the tension is 8.00 N, we can use the same wave speed formula:
Wave speed (v) = √(tension (T) / linear mass density (μ))
Given tension (T) = 8.00 N, we substitute the values into the formula:
v = √(8.00 N / 0.012 kg/m)
v = √666.67 m^2/s^2
v ≈ 25.81 m/s
Therefore, when the tension in the string is 8.00 N, the wave speed in the string is approximately 25.81 m/s.
a) To find the required tension in the string, we can use the equation:
T = μv²
where T is the tension in the string, μ is the linear mass density of the string (mass per unit length), and v is the speed of the wave.
In this case, the length of the string is given as 5.00 m and the total mass is given as 60 grams. However, we need to convert the mass to kilograms and the length to kilograms per meter (kg/m).
Given:
Length of string = 5.00 m
Total mass of string = 60 grams = 0.06 kg
To find the linear mass density (μ), we can divide the total mass by the length of the string:
μ = (Total mass) / (Length of string)
= 0.06 kg / 5.00 m
Now, we can substitute the values of μ and v into the equation to find the tension (T):
T = μv²
T = (0.06 kg / 5.00 m) * (50.0 m/s)²
Now, let's calculate the value of T:
T = (0.06 kg / 5.00 m) * (50.0 m/s)²
T ≈ 1.2 kg/s² * 2500 m²/s²
T ≈ 3000 N
Therefore, the required tension in the string is approximately 3000 N.
b) To calculate the wave speed in the string, we can rearrange the equation from part a) as follows:
v = √(T / μ)
Given:
Tension in the string (T) = 8.00 N
Linear mass density (μ) = 0.06 kg / 5.00 m
Now, we can substitute the values of T and μ into the equation to find the wave speed (v):
v = √(T / μ)
v = √(8.00 N / (0.06 kg / 5.00 m))
Now, let's calculate the value of v:
v = √(8.00 N / (0.06 kg / 5.00 m))
v = √(8.00 N / (0.06 kg / 5.00 m))
v = √(8.00 N * (5.00 m / 0.06 kg))
v = √(40.0 N * m / 0.06 kg)
v ≈ √(666.7 N * m / kg)
v ≈ 25.8 m/s
Therefore, the wave speed in the string, when the tension is 8.00 N, is approximately 25.8 m/s.