recall and apply the principle of conservation of energy

Energy Conservation

Learning Objective

Recall and apply the principle of conservation of energy to a variety of physical situations.

1. Statement of the Principle

The total energy of an isolated system remains constant; it can be transferred between forms or between objects, but it cannot be created or destroyed.

Mathematically, for an isolated system:

$$\Delta E_{\text{total}} = 0$$

or equivalently

$$E_{\text{initial}} = E_{\text{final}}$$

2. Forms of Energy Relevant to A‑Level Physics

• Kinetic energy, $K = \frac{1}{2}mv^{2}$
• Gravitational potential energy, $U_g = mgh$ (near Earth’s surface)
• Elastic potential energy, $U_e = \frac{1}{2}kx^{2}$
• Thermal energy (internal energy), $U_{\text{th}}$
• Other forms (chemical, nuclear) are usually outside the A‑Level syllabus but obey the same principle.

3. Work–Energy Theorem

The work done on an object equals the change in its kinetic energy:

$$W_{\text{net}} = \Delta K = K_{\text{final}} - K_{\text{initial}}$$

When only conservative forces act, the work done can be expressed as the negative change in potential energy, leading to the mechanical energy conservation equation:

$$K_{\text{i}} + U_{\text{i}} = K_{\text{f}} + U_{\text{f}}$$

4. Applying Conservation of Energy

Follow these steps when solving a problem:

1. Identify the system and confirm it is isolated (or account for non‑conservative work).
2. List all relevant forms of energy at the start and at the end.
3. Write the conservation equation $E_{\text{initial}} = E_{\text{final}}$.
4. Include any work done by non‑conservative forces (e.g., friction) as $W_{\text{nc}}$ on the appropriate side of the equation.
5. Solve for the unknown quantity.

5. Example Problems

Example 1 – Falling Object

A 2.0 kg ball is dropped from a height of 5.0 m. Neglect air resistance. Find its speed just before it hits the ground.

Solution:

$$\begin{aligned} &\text{Initial energy: }E_i = K_i + U_i = 0 + mgh = (2.0)(9.8)(5.0) = 98\ \text{J}\\ &\text{Final energy: }E_f = K_f + U_f = \frac{1}{2}mv^2 + 0\\ &\text{Conservation: }E_i = E_f \\ &\frac{1}{2}mv^2 = 98\ \text{J} \\ &v = \sqrt{\frac{2\times98}{2.0}} = \sqrt{98} \approx 9.9\ \text{m s}^{-1} \end{aligned}$$

Example 2 – Spring‑Loaded Cart

A cart of mass 0.5 kg is attached to a horizontal spring (constant $k = 200\ \text{N m}^{-1}$) compressed by 0.15 m. The cart is released on a frictionless track. Determine the speed of the cart when the spring returns to its natural length.

Solution:

$$\begin{aligned} &U_{\text{spring}} = \frac{1}{2}kx^2 = \frac{1}{2}(200)(0.15)^2 = 2.25\ \text{J}\\ &K_{\text{final}} = \frac{1}{2}mv^2 = U_{\text{spring}}\\ &\frac{1}{2}(0.5)v^2 = 2.25\\ &v^2 = \frac{2\times2.25}{0.5}=9\\ &v = 3.0\ \text{m s}^{-1} \end{aligned}$$

6. Common Misconceptions

• Energy is “used up”. Energy changes form; the total amount stays the same in an isolated system.
• Only kinetic energy matters. Potential energy must be included whenever forces are conservative.
• Friction creates energy. Friction converts mechanical energy into thermal energy, which must be accounted for as a loss from the mechanical system.

7. Practice Questions

Question	Key Concepts	Answer (for teacher)
A 1.5 kg block slides down a frictionless 30° incline from a height of 2.0 m. Find its speed at the bottom.	Gravitational PE → KE, $mgh = \frac12 mv^2$	$v = \sqrt{2gh} = \sqrt{2\times9.8\times2.0} \approx 6.3\ \text{m s}^{-1}$
A 0.8 kg pendulum bob is released from a horizontal position. What is its speed at the lowest point? (Neglect air resistance.)	PE loss = KE gain, $mgh = \frac12 mv^2$ with $h = L(1-\cos\theta)$	$v = \sqrt{2gL}$ (for $\theta = 90^\circ$, $h = L$)
A 3.0 kg cart moving at 4.0 m s⁻¹ encounters a rough patch that does 12 J of negative work. What is its speed after the patch?	Work–energy theorem, $W_{\text{nc}} = \Delta K$	$\frac12 m v_f^2 = \frac12 m v_i^2 + W_{\text{nc}}$ → $v_f = \sqrt{v_i^2 + 2W_{\text{nc}}/m} \approx 2.9\ \text{m s}^{-1}$

8. Suggested Diagram

9. Summary Checklist

• Identify all forms of energy present.
• Determine whether the system is isolated or if non‑conservative work is present.
• Write the conservation equation correctly, placing $W_{\text{nc}}$ on the appropriate side.
• Solve algebraically for the required quantity.
• Check units and reasonableness of the answer.