Energy conservation problems — when to use work-energy theorem vs conservation

Question

A block of mass $2$ kg slides down a rough incline of height $5$ m. The coefficient of kinetic friction is $0.3$ and the incline angle is $30°$ . Find the speed at the bottom. Which method should we use — work-energy theorem or mechanical energy conservation?

(JEE Main 2024 pattern)

Solution — Step by Step

Work-energy theorem says: $W_{\text{net}} = \Delta KE$ . It works always — friction or no friction.

Mechanical energy conservation says: $KE_i + PE_i = KE_f + PE_f$ . This works ONLY when there are no non-conservative forces (no friction, no air resistance).

Here we have friction, so pure energy conservation won’t work. We use the work-energy theorem (or equivalently, energy conservation with a friction work term).

Length of incline: $l = h/\sin\theta = 5/\sin 30° = 10$ m.

Work by gravity: $W_g = mgh = 2 \times 10 \times 5 = 100$ J

Normal force: $N = mg\cos\theta = 2 \times 10 \times \cos 30° = 10\sqrt{3}$ N

Friction force: $f = \mu N = 0.3 \times 10\sqrt{3} = 3\sqrt{3}$ N

Work by friction: $W_f = -f \times l = -3\sqrt{3} \times 10 = -30\sqrt{3} \approx -52$ J (negative because friction opposes motion)

W_{\text{net}} = \Delta KE = \frac{1}{2}mv^2 - 0

100 - 30\sqrt{3} = \frac{1}{2}(2)v^2

v^2 = 100 - 30\sqrt{3} \approx 100 - 52 = 48

v \approx \mathbf{6.93 \text{ m/s}}

Why This Works

The decision tree for energy methods is straightforward:

graph TD
    A["Energy Problem"] --> B{"Non-conservative forces<br/>present? (friction, air drag)"}
    B -->|"No"| C["Use Conservation of<br/>Mechanical Energy<br/>KE₁ + PE₁ = KE₂ + PE₂"]
    B -->|"Yes"| D{"Know the force<br/>explicitly?"}
    D -->|"Yes"| E["Work-Energy Theorem<br/>W_net = ΔKE"]
    D -->|"No, but know<br/>energy lost"| F["Modified Conservation<br/>KE₁ + PE₁ = KE₂ + PE₂ + E_lost"]
    C --> G["Faster: no force<br/>calculation needed"]
    E --> H["More general:<br/>works with any forces"]

Energy conservation is a shortcut — it lets you skip the force analysis entirely for conservative systems. The work-energy theorem is more general but requires calculating work done by each force. In problems with friction, we often combine both: use energy conservation and add a “friction loss” term.

Alternative Method — Modified Energy Conservation

Instead of calculating work by each force separately, write:

mgh = \frac{1}{2}mv^2 + \mu mg\cos\theta \times l

This directly says: initial PE = final KE + energy lost to friction. Same answer, slightly fewer steps.

For JEE numericals, the modified conservation approach is fastest when friction is the only non-conservative force. You avoid computing $W_g$ and $W_f$ separately. But if there are multiple non-conservative forces (e.g., applied force + friction), the work-energy theorem is cleaner.

Common Mistake

The classic error: using $KE_i + PE_i = KE_f + PE_f$ even when friction is present. This gives the wrong (higher) speed because you’ve ignored energy dissipated as heat. If your answer for “speed at the bottom of a rough incline” equals $\sqrt{2gh}$ , you’ve definitely missed the friction term — that formula only works for a smooth incline.

Question

Solution — Step by Step

Why This Works

Alternative Method — Modified Energy Conservation

Common Mistake

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