A stretched string of length 168 cm vibrates in its nth mode. The separation distance from node to antinode is 21 cm . What is the value of n?
To find the value of n, we need to understand the concept of nodes and antinodes in vibrating strings.
In the vibration of a string, a node is a point on the string that remains stationary, while an antinode is a point on the string that oscillates with maximum amplitude. The distance between a node and an adjacent antinode is equal to half a wavelength.
In this problem, we are given that the separation distance from a node to an antinode is 21 cm. Therefore, this distance corresponds to half a wavelength (λ/2).
We are also given that the length of the string is 168 cm. The length of a vibrating string is related to the wavelength by the formula:
Length of string = (n * λ) / 2,
where n is the mode of vibration (an integer) and λ is the wavelength.
Since we know that the separation distance from a node to an antinode is equal to half a wavelength, we can rewrite the formula as:
Length of string = n * (λ/2) = n * 21 cm.
Substituting the given length of the string (168 cm) into the equation:
168 cm = n * 21 cm.
To find the value of n, we can simply divide both sides of the equation by 21 cm:
168 cm / 21 cm = n.
Simplifying the equation gives:
8 = n.
Therefore, the value of n is 8.
To find the value of n, we can use the formula for the separation distance between a node and an antinode in a vibrating string:
d = (λ/2) * n
Where:
d is the separation distance between a node and an antinode,
λ is the wavelength,
n is the mode of vibration.
In this case, the separation distance is given as 21 cm and the length of the string is 168 cm.
d = 21 cm
Total length of string (L) = 168 cm
λ = 2L/n
Substituting the values into the equation, we can solve for n:
21 cm = (2 * 168 cm) / n
Multiplying both sides by n and dividing both sides by 21 cm, we get:
n = (2 * 168 cm) / 21 cm
Simplifying further:
n = 16
Therefore, the value of n is 16.