A 1m long string having mass of 4 gm is stretched by a force of 78.4N. calculate the frequency of the fundamental mode of vibration and the mass by which the string was stretched at its one end.
To calculate the frequency of the fundamental mode of vibration and the mass by which the string was stretched at its one end, we need to use the formula for the frequency of a stretched string and the equation for the mass per unit length of the string.
Let's start by calculating the mass per unit length of the string:
Mass per unit length (μ) = mass of the string (m) / length of the string (L)
Given: mass of the string (m) = 4 gm (0.004 kg)
length of the string (L) = 1 m
μ = 0.004 kg / 1 m = 0.004 kg/m
Next, we can calculate the tension in the string (T) using the force applied:
Tension (T) = force applied (F)
Given: force applied (F) = 78.4 N
T = 78.4 N
Now, we can use the formula to calculate the fundamental frequency of the string:
Fundamental frequency (f) = (1 / 2L) * √(T / μ)
f = (1 / 2 * 1) * √(78.4 / 0.004)
f = (1 / 2) * √(78.4 / 0.004)
Now we can calculate the frequency:
f = (1 / 2) * √(19600) = 70 Hz
Therefore, the frequency of the fundamental mode of vibration is 70 Hz.
To calculate the mass by which the string was stretched at its one end, we need to use the concept that the mass of the stretched string is equal to the mass of the string plus the mass by which it was stretched. Let's denote the mass by which it was stretched as 'Δm'.
Mass of stretched string (m_stretched) = mass of the string (m) + mass by which it was stretched (Δm)
m_stretched = 0.004 kg + Δm
To find Δm, we can use the formula for tension in a string:
Tension in the string (T) = mass per unit length (μ) * length of the string (L) * acceleration due to gravity (g)
Given: mass per unit length (μ) = 0.004 kg/m
length of the string (L) = 1 m
acceleration due to gravity (g) ≈ 9.8 m/s^2
T = μ * L * g
78.4 N = 0.004 kg/m * 1 m * 9.8 m/s^2
Now we can solve for Δm:
Δm = T / (L * g) - m
= 78.4 N / (1 m * 9.8 m/s^2) - 0.004 kg
= 8 kg
Therefore, the mass by which the string was stretched at its one end is 8 grams.