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derive-time-period-of-pendulum

๐Ÿš€ The simple pendulum is a classic example of harmonic motion. It consists of a mass (or bob) attached to a string or rod of fixed length that swings back and forth under the influence of gravity. The time period of a pendulum is the time taken for one complete cycle of motion, which is determined by the length of the pendulum and the acceleration due to gravity. This relationship can be derived using the principles of physics and mathematics.

Theory Explanation

Understanding the Forces on the Pendulum

When the pendulum is displaced from its equilibrium position, gravitational force acts on the bob, causing it to accelerate back towards the equilibrium position. The restoring force is proportional to the sine of the angle of displacement, which for small angles can be approximated as the angle itself (in radians). This leads to simple harmonic motion.

\[ F = -mg \sin(\theta) \approx -mg\theta \]
Setting Up the Equation of Motion

Using Newton's second law, we can set up the equation of motion for the pendulum. The net force acting on the bob is equal to the mass times its acceleration. This leads to the differential equation that describes the motion of the pendulum.

\[ m \frac{d^2\theta}{dt^2} = -mg\sin(\theta) \approx -mg\theta \]
Solving the Differential Equation

The differential equation can be solved to find the time period of the pendulum. The solution reveals that the motion is periodic and the time period depends on the length of the pendulum and the acceleration due to gravity.

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

Key Points

  • ๐ŸŽฏ The time period of a simple pendulum is independent of its mass.
  • ๐ŸŽฏ The time period increases with the length of the pendulum.
  • ๐ŸŽฏ The time period is affected by the acceleration due to gravity.

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Examples:💡

Calculate the time period of a pendulum of length 2 meters.

Solution:

Step 1: Identify the length of the pendulum (L = 2 m) and the acceleration due to gravity (g = 9.81 m/sยฒ).

Step 2: Use the formula for the time period: T = 2ฯ€โˆš(L/g).

Step 3: Substituting the values: T = 2ฯ€โˆš(2/9.81).

Step 4: Calculate T: T โ‰ˆ 2.83 seconds.

Common Mistakes

  • Mistake: Confusing the time period with the frequency of the pendulum.

    Correction: Remember that frequency (f) is the reciprocal of the time period (T): f = 1/T.

  • Mistake: Assuming the time period is the same for all pendulum lengths.

    Correction: Understand that the time period increases with the length of the pendulum.

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