Loading history...
Mechanical Energy
Definition
Mechanical energy is the sum of kinetic energy and potential energy in a system.\[E_m = E_k + E_g = \frac{1}{2}mv^2 + mgh\]
Where:
- E_k = kinetic energy (energy of motion)
- E_g = potential energy (stored energy due to position)
Types of Mechanical Energy
1. Kinetic Energy (Ek)
Energy of motion:
\[E_k = \frac{1}{2}mv^2\]
Characteristics:
- Zero when object is at rest
- Increases with velocity squared
- Always positive
2. Gravitational Potential Energy (Eg)
Energy stored due to height:
\[E_g = mgh\]
Characteristics:
- Relative to chosen reference level
- Higher position = more PE
- Zero at reference level
3. Elastic Potential Energy (Ee)
Energy stored in compressed/stretched materials:
\[E_e = \frac{1}{2}kx^2\]
Conservation of Mechanical Energy
The Principle
In the absence of friction and other dissipative forces, the total mechanical energy of a system remains constant.\[E_{\text{total}} = E_k + E_g = \text{constant}\]
Mathematical Expression
At any two points in motion:
\[E_{k1} + E_{g1} = E_{k2} + E_{g2}\]
\[\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2\]
Example: A 2 kg ball dropped from 20 m height (g ≈ 10 m/s²)
Initial state (at rest, 20 m high):
- \(E_{k0} = 0 J\)
- \(E_{p0} = 2 \cdot 10 \cdot 20 = 400 J\)
- \(E_{m0} = 0 + 400 = 400 J \)
- \(E_{p} = 2 \cdot 10 \cdot 10 = 200 J\)
- \(E_{k} = 400 - 200 = 200 J\)
- \(E_m = 200 + 200 = 400 J \)
- \(E_{p} = 0 J\)
- \(E_{k} = 400 J\)
- \(E_m = 0 + 400 = 400 J \)