——— 💥 Work & Energy ———

——— 🚠 Work ———

~ 📖 Definition ~

~ 🔄 Varying Force ~

If the force changes with position:

$$W = \int_0^d F(x)\, dx$$

~ 🧪 Example ~

A 1 kg object is pushed along a straight surface. The applied force decreases linearly from 15 N to 0 N over a distance of d = 0.6 m.

1) Expressing Force Function \( F(x) \):

(Force \( F(x) \) is defined based on how it changes over the distance.)

2) Work Calculation:

$$W = \int_0^d 15\left(1 - \frac{x}{d}\right)\, dx$$
$$= 15 \int_0^d \left(1 - \frac{x}{d}\right) dx$$
$$= 15 \left[ x - \frac{x^2}{2d} \right]_0^d$$
$$= 15 \left(d - \frac{d}{2}\right) = 15 \cdot \frac{d}{2}$$

Plugging in \( d = 0.6 \, m \):

$$W = 15 \cdot 0.3 = 4.5 \, J$$

✅ Final Answer: Total work done on the object is 4.5 Joules.

——— ⚡ Energy ———

~ 📖 Definition ~

Energy is the result of work. When work is done on a system, energy is transferred to it. Measured in Joules (J).

~ 🚀 Kinetic Energy ~

$$KE = \frac{1}{2}mv^2$$
$$W_{net} = KE_f - KE_i$$

~ 🏔️ Potential Energy ~

Gravitational:

$$PE = mgh$$

Spring (Elastic):

——— ♻️ Conservation of Mechanical Energy ———

~ 📖 Definition ~

If no external forces (like friction or air resistance) take energy away from the system, then the total mechanical energy remains constant.

$$E_{\text{mechanical}} = KE + PE$$

The Core Principle:

$$E_{\text{initial}} = E_{\text{final}} \quad \Rightarrow \quad KE_i + PE_i = KE_f + PE_f$$

This means: If you push, throw, lift, or drop something — as long as no energy is lost (for example, due to heat, friction, or air resistance) — its total mechanical energy stays the same. The energy just shifts between forms, like kinetic and potential.