Question

The speed \( v \) of a wave on a string depends on the tension \( F \) in the string and the mass per unit length \( m/L \) of the string. If it is known that [F] = [ML][T]–2, the values of the constants \( a \) and \( b \) in the following equation for the speed of a wave on a string are: \[ v = (\text{constant}) F^a \left( \frac{m}{L} \right)^b \]

• A. \( a = \frac{1}{2}, b = \frac{1}{2} \)
• B. \( a = 2, b = -1 \)
• C. \( a = \frac{1}{2}, b = -1 \)
• D. \( a = 1, b = \frac{1}{2} \)

Hint:

The speed of a wave on a string depends on the tension and mass per unit length, with the speed being proportional to the square root of the tension and inversely proportional to the square root of the mass per unit length.

Answer 1

The Correct Option is A

The speed of the wave on the string is given by: \[ v = \sqrt{\frac{F}{\mu}} \] Where \( \mu = \frac{m}{L} \) is the mass per unit length of the string. Therefore, the speed is proportional to the square root of the tension \( F \) and the inverse square root of the mass per unit length. Thus, \( a = \frac{1}{2} \) and \( b = \frac{1}{2} \).