To determine the relationship between the separation d of the clamps and the tension T in the cord, we can consider the standing-wave pattern formed by the vibrating cord. In a standing wave, there are nodes and antinodes. Nodes are points of minimum displacement, while antinodes are points of maximum displacement.
In the simplest standing-wave vibration mode, there is only one antinode in the center of the cord and two nodes at the clamps. This means that the distance between two consecutive nodes or antinodes is half the wavelength (λ/2) of the wave.
The wavelength (λ) of a wave can be calculated using the formula:
λ = v/f
Where:
- λ is the wavelength
- v is the velocity of the wave
- f is the frequency of the wave
Since the cord is vibrating in resonance, the wavelength of the wave on the cord is equal to twice the length of the cord (2L), where L is the length between the clamps.
Therefore, λ = 2L
Now, let's consider the tension in the cord. The tension (T) in the cord creates a restoring force that allows the cord to vibrate. The relationship between the tension and the mass per length (μ) of the cord is given by the formula:
T = μ * g * L
Where:
- T is the tension
- μ is the mass per length
- g is the acceleration due to gravity
- L is the length of the cord
Finally, to determine the relationship between the separation d of the clamps and the tension T in the cord, we need to consider that the separation d is equal to the length of the cord L. Therefore, the relationship can be expressed as:
d = L
So, the separation d of the clamps is equal to the length of the cord L, and the tension T in the cord can be calculated using the formula T = μ * g * L, where μ is the mass per length of the cord.