What are the two laws of pendulum?
1 Answer
The two key laws governing the motion of a pendulum are:
### 1. **The Law of Isochronism**
- **Definition**: This law states that the time period of a simple pendulum (the time it takes for the pendulum to complete one full swing back and forth) is constant and does not depend on the amplitude of the swing, as long as the amplitude is small.
- **Explanation**: When a pendulum swings through a small angle, its period remains nearly constant. This means that the time it takes to complete one oscillation (or cycle) is the same, regardless of how far it swings. This is true as long as the angle (or displacement) from the equilibrium position is small (usually less than 15°).
- **Formula**: The time period \( T \) for a simple pendulum of length \( L \) and acceleration due to gravity \( g \) is given by:
\[
T = 2\pi \sqrt{\frac{L}{g}}
\]
In this equation:
- \( T \) is the period (time for one complete swing),
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.
This law assumes that the air resistance and friction are negligible.
### 2. **The Law of Conservation of Mechanical Energy**
- **Definition**: This law states that the total mechanical energy (the sum of kinetic and potential energy) of a pendulum remains constant throughout its motion, provided there are no non-conservative forces (like air resistance or friction) acting on it.
- **Explanation**: As the pendulum swings, its energy continuously converts between kinetic energy (when it is moving fastest at the lowest point) and potential energy (when it is at its highest point, where it momentarily stops before reversing direction). However, the total mechanical energy remains constant in the absence of friction or other energy losses.
- At the highest points of the swing, the pendulum has maximum potential energy and zero kinetic energy.
- At the lowest point, where the pendulum is moving fastest, it has maximum kinetic energy and minimum potential energy.
The conservation of mechanical energy can be mathematically expressed as:
\[
E = K + U
\]
where:
- \( E \) is the total mechanical energy,
- \( K \) is the kinetic energy, and
- \( U \) is the potential energy.
Because the total energy remains constant, any increase in one form (e.g., kinetic) is balanced by a decrease in the other form (e.g., potential).
### Summary:
- **Isochronism**: The period of a pendulum is independent of the amplitude for small angles.
- **Conservation of Mechanical Energy**: The total energy of a pendulum is conserved, converting between kinetic and potential energy as it swings.
### 1. **The Law of Isochronism**
- **Definition**: This law states that the time period of a simple pendulum (the time it takes for the pendulum to complete one full swing back and forth) is constant and does not depend on the amplitude of the swing, as long as the amplitude is small.
- **Explanation**: When a pendulum swings through a small angle, its period remains nearly constant. This means that the time it takes to complete one oscillation (or cycle) is the same, regardless of how far it swings. This is true as long as the angle (or displacement) from the equilibrium position is small (usually less than 15°).
- **Formula**: The time period \( T \) for a simple pendulum of length \( L \) and acceleration due to gravity \( g \) is given by:
\[
T = 2\pi \sqrt{\frac{L}{g}}
\]
In this equation:
- \( T \) is the period (time for one complete swing),
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.
This law assumes that the air resistance and friction are negligible.
### 2. **The Law of Conservation of Mechanical Energy**
- **Definition**: This law states that the total mechanical energy (the sum of kinetic and potential energy) of a pendulum remains constant throughout its motion, provided there are no non-conservative forces (like air resistance or friction) acting on it.
- **Explanation**: As the pendulum swings, its energy continuously converts between kinetic energy (when it is moving fastest at the lowest point) and potential energy (when it is at its highest point, where it momentarily stops before reversing direction). However, the total mechanical energy remains constant in the absence of friction or other energy losses.
- At the highest points of the swing, the pendulum has maximum potential energy and zero kinetic energy.
- At the lowest point, where the pendulum is moving fastest, it has maximum kinetic energy and minimum potential energy.
The conservation of mechanical energy can be mathematically expressed as:
\[
E = K + U
\]
where:
- \( E \) is the total mechanical energy,
- \( K \) is the kinetic energy, and
- \( U \) is the potential energy.
Because the total energy remains constant, any increase in one form (e.g., kinetic) is balanced by a decrease in the other form (e.g., potential).
### Summary:
- **Isochronism**: The period of a pendulum is independent of the amplitude for small angles.
- **Conservation of Mechanical Energy**: The total energy of a pendulum is conserved, converting between kinetic and potential energy as it swings.