Pendulums

Formula for the Time Period of a Simple Pendulum

The time period of a simple pendulum is given by:

$ = Length of the pendulum

= Acceleration due to gravity

Step 1: Calculate the Original Time Period

For a pendulum with mass and length , the time period is:

\) ).

Step 3: Compute the New Time Period

The time period of the new pendulum is:

\) :

\)

According to the question:

\) :

\) $

Conclusion

The value of is .

To solve the problem, we need to calculate the ratio of the speed of the bob at point P to its initial speed . Let's follow the steps:-

Step 1: Apply Conservation of Energy

The total mechanical energy at the start and at point P can be equated since only conservative forces are acting:

After canceling mass and rearranging, we get:

Step 2: Apply Dynamics at Point P

For the string to become slack at point P, the tension in the string becomes zero:

At point P, centripetal force is provided by the component of the gravitational force:

Simplifying:

Step 3: Relate Both Equations

Substitute the expression for from the centripetal force equation into the energy equation:

Simplify and solve for :

Factor out :

Thus, the ratio is:

Noticing that the additional trigonometric simplification aligns with: