Alright, let's imagine a metal spring, just like the ones you see in a mechanical pencil or a trampoline. You squish it or stretch it, and what happens? It springs back to its original shape! When we do this, we're storing energy in the spring, called elastic potential energy (E_H). Just like a rubber band shot across the room, this energy is released when the force is removed.

A scientist named Hooke found that when we apply a small load to a spring, it extends in proportion to the load applied. We call this Hooke’s law and express it as

F ∝ Δx or F = kΔx

Here

• F is the force
• Δx is the extension of the spring
• k is the spring constant

Imagine a graph where the x-axis is Δx and the y-axis is F. The graph will be a straight line going through the origin, and its slope will be equal to the spring constant, k.

The work done in stretching the spring, or the elastic potential energy (E_H), is given by the area of the triangle under the F – Δx graph, which is

E_H = \(\frac12\) F_max × Δx or E_H = \(\frac12\)k (Δx)²

This sounds a bit complex, so let's dive into an example to make it clearer!

Example I 🎒

A spring, with a spring constant 48 N/m, is extended by 0.40 m. How much elastic potential energy is stored in the spring?

The energy stored can be calculated by the formula E_H = \(\frac12\) kx²

Energy stored = \(\frac12\) × 48 × (0.40)² = 3.8 J

(Imagine if you have a toy gun with this spring! You could launch a foam bullet with 3.8 Joules of energy!)

Now let's move to efficiency. In the real world, when we transfer energy (like turning potential energy into kinetic energy), some of it gets lost due to friction, or gets stored as other types of energy. We can measure these losses with efficiency.

Efficiency is basically how well we convert input energy into useful output energy or power. It can be calculated by

Efficiency = (useful work out/total energy in) or (useful power output/total power input)

Example II 🚴‍♀️

A cyclist rides up a 50 m high hill in 200 s. The average power developed by the cyclist is 270 W, and the mass of the cyclist and the bicycle is 85 kg. What's the efficiency with which the work done by the cyclist is transferred to the gravitational potential energy?

Example III 🚗

An electric car of mass 1600 kg accelerating on a horizontal road converts 65% of the electrochemical energy stored in the battery to kinetic energy. What is the energy transferred from the battery when the car accelerates from rest to a speed of 50 km/h?

Example IV 🏀

A ball is dropped from a fixed height of 1.00 m so that it bounces several times. The subsequent heights that it reaches are measured after each successive bounce. Calculate the energy of the ball on each bounce and hence calculate the efficiency of the ball when it bounces.

Remember to keep these concepts in mind as you move forward in your physics journey!