Waves And Oscillations

From force law
where is force constant or spring constant of spring and is the extension or compression in spring.

or
or
If spring is divided in two parts, then


or
Hence, spring constant will be doubled.

Key Idea: Total energy of a particle executing simple harmonic motion is obtained by summing its potential and kinetic energies.
Potential energy of particle in SHM

or
or ?(i)
Kinetic energy of particle in SHM

or ?(ii)
Hence, total energy




Thus, it is obvious that total energy of particle executing simple harmonic motion depends on amplitude (A) and period (T). and are correct

Key Idea : Equating the given equation with general equation of . The given equation is written as
....(i)
The general equation of simple harmonic motion is written as
....(ii)
Equating Eqs. (i) and (ii), we get

Maximum velocity,

In order to find the time taken by the particle from-12.5 cm to
+ 12.5 cm on either side of mean position, we will find the
time taken by particle to go from x = - 12. 5 cm to x = 0 and to
go from x = 0 to x = + 12.5 cm.
Let the equation of motion be x
First, the particle moves from x = -125 cm to x = 0




Similarly to go from x = 0 to x = 12. 5 cm

Total time taken from x = -12.5 cm to x = 12.5 cm

To find the velocity of the bob at its mean position, we can use the principle of conservation of mechanical energy. The potential energy (PE) at the maximum height is converted into kinetic energy (KE) at the mean position.
Step-by-Step Solution:|
1. Calculate the Potential Energy (PE) at the maximum height:
The potential energy at the height is given by:

where:
- is the mass of the bob
- is the acceleration due to gravity ( )
- is the height ( )
2. Convert the Potential Energy (PE) into Kinetic Energy (KE) at the mean position:
At the mean position, the height is zero, so all the potential energy will be converted into kinetic energy. The kinetic energy is given by:

where:
- is the velocity of the bob at the mean position
3. Set PE equal to KE:

Notice that the mass cancels out from both sides of the equation:

4. Solve for :


5. Substitute the known values:



The velocity of the bob at its mean position is .

The simple pendulum at angular amplitude is shown in the figure. Maximum tension in the string is

When bob of the pendulum comes from to , it covers a vertidal distance

Also during to , potential energy of bob converts into kinetic energy ie,

Thus, using Eqs. (i), (ii) and (iii), we obtain

Velocity of SHM
...(i)
Similarly, ...(ii)
Dividing E (ii) by E (i), we get


or

Let x be the point where KE = PE

The correct option is (B):