Pendulums

Concept:

Step 1: Time taken.
Larger period pendulum:

Step 2: Angular frequencies.


Step 3: Phase difference.

Step 1: Understanding the Concept:
Period of simple pendulum: . So .

Step 2: Detailed Explanation:



Step 3: Final Answer:
The new period is seconds.

Concept:
Physics - Simple Pendulum Dynamics.
The time period is given by . Since frequency , the relationship is .
Step 1: Establish the ratio between frequency and length.
Based on the formula , we can write the ratio for two different lengths: $ L_2 = 4L_1 f_1 = f $

Step 1: Understanding the Concept:
In Simple Harmonic Motion (SHM), the acceleration of a particle is directly proportional to its displacement but in the opposite direction. The maximum acceleration occurs at the extreme positions (maximum displacement, which is the amplitude). The relationships between time period, angular frequency, acceleration, and amplitude dictate the dynamics of the system.
Step 2: Key Formula or Approach:
1. Maximum acceleration:
2. Time period:
Where is angular frequency and is amplitude.
We will find the new angular frequency when is doubled, and then find the new time period .
Step 3: Detailed Explanation:
Let the initial parameters be , , , and .


The problem states the new acceleration is doubled while keeping amplitude constant. This implies the maximum acceleration is doubled.
Let the new parameters be , , , and .
Given:
(amplitude is constant)

Using the acceleration formula for the new state:

Substitute the given relationships:

Since we know , substitute this in:

Cancel from both sides (since ):

Take the square root:

Now, find the new time period :

Substitute :

We know the initial time period is , so we can substitute :

Step 4: Final Answer:
The new time period is .