Two children are sending signals along a cord of total mass 0.50 kg tied between tin cans with a tension of 35 N. It takes the vibrations in the string 0.50 s to go from one child to the other. How far apart are the children?
To find the distance between the children, we can use the wave speed equation:
Wave speed = frequency × wavelength
In this case, the wave speed is equal to the speed of the vibrations in the string, which is given by the equation:
Wave speed = √(tension / linear mass density)
Where the linear mass density is equal to the total mass divided by the length of the cord:
Linear mass density = total mass / length
Let's calculate each step to find the distance between the children:
1. First, let's calculate the linear mass density:
Linear mass density = total mass / length
Given that the total mass is 0.50 kg, we need to find the length.
2. Next, let's calculate the wave speed:
Wave speed = √(tension / linear mass density)
The tension is given as 35 N.
3. Now, let's calculate the wavelength:
Wavelength = wave speed / frequency
The frequency is equal to the inverse of the time it takes for the vibrations to go from one child to the other.
4. Finally, let's find the distance:
The distance between the children is equal to half the wavelength because the vibrations travel from one child to the other and then back again.
To determine the distance between the children, we can use the wave speed equation:
Wave speed = Frequency × Wavelength
In this case, the frequency is the reciprocal of the time it takes for the vibration to travel from one child to the other:
Frequency = 1 / Time
Now, we need to find the wavelength. The wavelength is defined as the distance between two successive identical points on a wave. In this case, it is the distance between the two children.
We can rearrange the wave speed equation to solve for the wavelength:
Wavelength = Wave speed / Frequency
First, let's calculate the frequency:
Frequency = 1 / Time
= 1 / 0.50 s
= 2 Hz
Now, we can calculate the wavelength:
Wavelength = Wave speed / Frequency
To find the wave speed, we'll use the formula:
Wave speed = √(Tension / Linear density)
where Tension is the tension in the cord and Linear density is the mass per unit length.
In this case, the total mass of the cord is given as 0.50 kg. Since we don't know the length of the cord, we can't calculate the linear density directly. However, we can calculate the mass per unit length by dividing the total mass by the length of the cord. Let's denote the length of the cord as L:
Mass per unit length = Total mass / Length
= 0.50 kg / L
Now we can substitute this into the wave speed equation:
Wave speed = √(Tension / Linear density)
= √(35 N / (0.50 kg / L))
= √(35 N * (L / 0.50 kg))
= √(70 L) m/s
Finally, we can substitute the wave speed and frequency into the wavelength equation:
Wavelength = Wave speed / Frequency
= (√(70 L) m/s) / (2 Hz)
= √(70 L / 4) m
= √(17.5 L) m
So, the distance between the children is given by the wavelength, which is √(17.5 L), where L is the length of the cord.