91Ó°ÊÓ

Mechanical energy is a fundamental concept in physics that represents the sum of kinetic and potential energies in a system. For a pendulum, this total mechanical energy is vital in understanding its motion.
The mechanical energy of the pendulum remains constant as it swings - if air resistance and friction are ignored.
When the pendulum is at its lowest point, all of its mechanical energy is in the form of kinetic energy since it is at zero potential energy. As it swings upward, some mechanical energy is converted into potential energy.

• Mechanical Energy = Kinetic Energy + Potential Energy

For example, at the very bottom of its swing, a stone in a pendulum might have a mechanical energy of 64 Joules, all of which is kinetic energy.

Kinetic energy is the energy of motion. In the context of a pendulum, kinetic energy is at its maximum when the stone is moving fastest, specifically at the lowest point of the swing.
The formula to calculate kinetic energy is: \[ KE = \frac{1}{2} m v^2 \]
Here, \(m\) is the mass, and \(v\) is the velocity.

• This means in our scenario, when the stone weighs 2 kg and is moving at 8 m/s, the kinetic energy is 64 Joules at the lowest point.
• Some kinetic energy gets converted to potential energy as the pendulum rises.

Understanding kinetic energy helps explain how speeds vary at different points.

Potential energy can be thought of as stored energy. For a pendulum, this is primarily gravitational potential energy, which depends on the height of the pendulum above its lowest point.
The formula to determine potential energy is: \[ PE = mgh \]
This takes into account the mass \(m\), gravitational acceleration \(g\), and height \(h\).

• At the lowest point, potential energy is zero because height is zero.
• At 60 degrees, the pendulum raises to a height, acquiring potential energy - calculated as 39.2 Joules in this case.

As the stone climbs higher, it gains more potential energy while losing kinetic energy.

The conservation of energy principle implies that the total mechanical energy remains constant if no external force acts on the system. Thus, as a pendulum swings, it continuously transforms kinetic energy into potential energy and vice versa, without any loss.
In a practical application, this allows us to calculate various properties of the pendulum's motion, like speed at different angles, using:

• Total Mechanical Energy = Initial Kinetic Energy = Kinetic Energy + Potential Energy at any point.

For example, at 60 degrees, knowing the potential energy allows us to compute remaining kinetic energy and thus the speed, confirming the energy conservation law. Over its swing, this preserves a mechanical energy of 64 Joules.