Wave Motion

To solve the problem of finding the amplitude and phase of the resultant wave formed by the superposition of two harmonic traveling waves, we need to analyze the given waves:

• The first wave is
• The second wave is

When two waves superimpose, they add up to create a resultant wave. The equation for the resultant wave can be written as:

Where is the amplitude and is the phase of the resultant wave. The resultant amplitude can be found using the formula for the superposition of two sinusoidal functions:

Where:

• : amplitude of the first wave.
• : amplitude of the second wave.
• : phase difference between the two waves.

By substituting the values, we get:

Since , substitute in the equation:

Thus, the amplitude .

Next, we find the phase using the formula:

Substituting the values, we get:

Since and :

Thus, .

Therefore, the amplitude and phase of the resultant wave are .

Correct Answer:

To solve this problem, we need to examine the relationship between the velocity of a transverse wave in a string and the string's physical properties. The velocity of a transverse wave on a string is given by the formula:

where:

• is the tension in the string.
• is the linear mass density of the string.

Since both strings are made from the same material and are under the same tension, the critical factor that affects the wave's velocity is the linear mass density . Linear mass density is defined as:

,

where is the mass of the string and is its length. For a string with a circular cross-section, can also be described in terms of its volume as:

where:

• is the material's density
• is the cross-sectional area

For a circular cross-section, the area is given by:

Therefore, the linear mass density becomes:

Since the tension is the same, the ratio of velocities is determined by the ratio of the square roots of the inverse of their linear densities:

For the first string with radius , the linear density is:

For the second string with radius , the linear density is:

Calculating the ratio:

Thus, the ratio is . Therefore, the correct answer is:

Option: 2

To find the velocity of the transverse wave given by the equation mm, we need to analyze the wave equation:

where:

• is the amplitude of the wave.
• is the wave number (measured in rad/m).
• is the angular frequency (measured in rad/s).
• is the displacement at position and time .

From the given equation, we identify:

• Amplitude, mm.
• Wave number, rad/mm rad/m (since there are 1000 mm in a meter).
• Angular frequency, rad/s.

The wave velocity is given by the formula:

Substituting the values found:

The solution here gives us the magnitude of the wave velocity as 30 m/s. However, the wave travels in the negative x-direction, as indicated by the positive sign in the wave equation ( ). This means the direction of propagation is negative.

Therefore, the velocity of the wave is -30 m/s, matching with the correct option -30 m/s.

Thus, the correct answer is -30 m/s.

Step 1: Analyze Statement 1.
The total kinetic energy of a system of particles is the sum of the kinetic energies of each particle. This matches Statement 1: Thus, Statement 1 is true.
Step 2: Analyze Statement 2.
The kinetic energy of a system can also be described as the kinetic energy of the center of mass (which is ) plus the kinetic energy due to the motion of particles relative to the center of mass. This matches Statement 2.
Step 3: Conclusion.
Both Statement 1 and Statement 2 are correct.
Final Answer: