Waves And Oscillations

In a complete cycle of SHM, potential energy varies for half the cycle and kinetic energy varies for the other half of the cycle. Thus, for a time period T, the potential energy varies for

Answer (d)

A spring of spring constant is attached to a rigid wall. A ball is placed on the horizontal table as shown in figure

If spring is compressed then potential energy When spring hit the ball, it transfer all its potential energy to kinetic energy of ball Given, and

- Option A: Stiff springs have a high value of spring constant , which represents the stiffness of the spring. This statement is correct. - Option B: Soft springs have a low value of spring constant , which indicates a less stiff spring. This statement is correct. - Option C: The spring constant is actually dependent on the material's elasticity and the dimensions of the spring. A spring with a different material or structure will have a different spring constant, so this statement is false. - Option D: For small oscillations, the spring follows simple harmonic motion (SHM) where the restoring force is directly proportional to the displacement. This statement is correct. - Option E: The period of oscillations of a spring is given by , so the period depends on the value of . This statement is correct. Thus, the false statement is (C), as the spring constant is indeed dependent on the elastic properties of the spring.

Concept:

Step 1: Given.


Step 2: Substitute .


Step 3: Time period.

Concept:
• Closed pipe: only odd harmonics
• Open pipe: all harmonics

Step 1: Third mode.

• Closed pipe 5th harmonic
• Open pipe 3rd harmonic

Step 2: Frequency ratio.

Step 1: Define the equations of motion for a particle in SHM.
Let the displacement of a particle in Simple Harmonic Motion (SHM) be represented by: where is the amplitude, is the angular frequency, and is the initial phase.

Step 2: Derive the expressions for velocity and acceleration.
Velocity is the first derivative of displacement with respect to time: Acceleration is the first derivative of velocity with respect to time:

Step 3: Express velocity and acceleration using the same trigonometric function to compare their phases.
We can use the trigonometric identities and . The velocity equation becomes: The phase of the velocity is . The acceleration equation becomes: The phase of the acceleration is .

Step 4: Calculate the phase difference.
The phase difference is the phase of acceleration minus the phase of velocity. A positive phase difference means acceleration leads velocity.

Final Answer: (C)

Step 1: Understanding the Concept:
When springs are connected in series, the same force (tension) acts on each spring, but their extensions add up. This arrangement can be replaced by a single equivalent spring with an effective spring constant. The time period of a mass-spring system depends on mass and this effective spring constant.

Step 2: Key Formula or Approach:
1. The equivalent spring constant ( ) for two springs ( ) in series is:

2. The time period ( ) of the oscillating mass is:


Step 3: Detailed Explanation:
Given spring constants: and .
Calculate the equivalent spring constant :

Find a common denominator:

Inverting both sides gives:

Now, substitute into the time period formula:




Step 4: Final Answer:
The time period of oscillation is .