Mada za sehemu hiiMotion In Straight LineMada 5
Definition: A simple pendulum is a small heavy object (called a bob) suspended from a fixed point by a light, inextensible, and non-elastic string such that it can swing freely in a vertical plane.
Concept: The pendulum oscillates back and forth under the action of gravity. The time it takes for one complete swing (to and fro) is called the period.
Let:
= Period of oscillation (in seconds) = Length of the pendulum (in meters) = Acceleration due to gravity (in )
Formula for the period of a simple pendulum:
To find acceleration due to gravity: Rearranging the formula above to solve for :
Step 1: Start with the period formula
Step 2: Divide both sides by
Step 3: Square both sides
Step 4: Solve for
Which simplifies to:
The acceleration due to gravity can be calculated by measuring the length of the pendulum and the period of oscillation, and using the formula:
This experiment is commonly done in schools and laboratories to determine the value of , typically found to be around .
- Period (T): The time taken by the pendulum bob to complete one full oscillation (to and fro).
- Angular displacement: The angle made between the string and the vertical axis when the bob is displaced to the maximum position.
- Amplitude: The maximum displacement of the bob from the mean position (equilibrium).
- Length of pendulum (l): The distance from the point of suspension to the center of gravity of the bob.
The period of a simple pendulum is given by the formula:
Derivation step-by-step:
Step 1: Start with the basic period formula
Step 2: Square both sides to eliminate the square root:
Step 3: Rearranging to express length (l):
If we plot a graph of l against T², the slope of the straight line will be:
And the y-intercept will be 0 since the equation is of the form .
- When the bob is raised to point B (or C), it possesses potential energy (P.E.).
- As it swings down to the lowest point O, the potential energy is converted into kinetic energy (K.E.).
- At points B and C: All energy is potential energy (P.E.).
- At point O: All energy is kinetic energy (K.E.).
By conservation of energy:
Energy loss in real-life: In a real pendulum, due to air resistance and internal friction, energy is gradually lost, and the amplitude decreases until the pendulum stops oscillating. However, in a vacuum, the pendulum would swing indefinitely since there's no energy loss.
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