A transverse wave travels with a speed of 20m/s INA strung under tension due to the weight of an object of mass 0.6 kg. The made required for a wave speed of 30m/s in the same string is

To determine the mass required for a wave speed of 30 m/s in the same string, we can start by using the relationship between wave speed, tension, and mass per unit length of the string. The wave speed \( v \) on a string is given by the formula:

Wave Speed Formula

The formula is:

v = √(T/μ)

Where:

• v = wave speed
• T = tension in the string
• μ = mass per unit length of the string

Finding the Initial Tension

Initially, we know that the wave speed is 20 m/s with a mass of 0.6 kg hanging from the string. The tension \( T \) in the string due to the weight of the object can be calculated using:

T = m * g

Where:

• m = mass (0.6 kg)
• g = acceleration due to gravity (approximately 9.81 m/s²)

Calculating the tension:

T = 0.6 kg * 9.81 m/s² = 5.886 N

Calculating Mass per Unit Length

Now, we can find the mass per unit length \( μ \) using the wave speed formula:

20 m/s = √(5.886 N / μ)

Squaring both sides gives:

400 = 5.886 / μ

Rearranging this to solve for \( μ \):

μ = 5.886 / 400 = 0.014715 kg/m

Finding the New Tension for 30 m/s

Next, we want to find the new tension required for a wave speed of 30 m/s. Using the same wave speed formula:

30 m/s = √(T_new / 0.014715 kg/m)

Squaring both sides results in:

900 = T_new / 0.014715

Rearranging gives:

T_new = 900 * 0.014715 = 13.2435 N

Calculating the Required Mass

To find the mass that will create this tension, we use the tension formula again:

T_new = m_new * g

Solving for \( m_new \):

m_new = T_new / g = 13.2435 N / 9.81 m/s²

Calculating this gives:

m_new ≈ 1.349 kg

Summary

To achieve a wave speed of 30 m/s in the same string, the mass required is approximately 1.349 kg. This demonstrates how tension and mass interact to influence wave speed in a string, highlighting the fundamental principles of wave mechanics.