PHSX 220 Physics I (w/ Calculus) · Unit 2: Applying Newton, statics, fluids, energy · on Exam 2
Ch 8 Potential energy and conservation of energy
Study notes for Potential energy and conservation of energy in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Conservative forces (gravity, springs) have potential energy: $U_g = mgy$ (any zero level you like), $U_s = \tfrac12 kx^2$. Their work is path-independent and $W_c = -\Delta U$.
- Conservation of mechanical energy when only conservative forces do work: $K_i + U_i = K_f + U_f$. With friction or other non-conservative forces: $K_i + U_i + W_{nc} = K_f + U_f$, where $W_{nc} = -f_k d$ for kinetic friction.
- Speed at the bottom of a frictionless track depends only on the height drop, not on the path: $v = \sqrt{2gh}$.
- Energy diagrams: the object moves where $E \ge U(x)$; turning points are where $E = U$; equilibrium where $dU/dx = 0$ (stable at a minimum of $U$).
- Energy is a scalar: no components, no directions. That is why it is often the easiest tool.
Key formulas
Potential energies
$$U_g = mgy,\qquad U_s = \tfrac12 kx^2,\qquad F_x = -\frac{dU}{dx}$$
Conservation with friction
$$\tfrac12 mv_i^2 + U_i - f_k d = \tfrac12 mv_f^2 + U_f$$
Worked example
A 0.50 kg block compresses a spring ($k = 800$ N/m) by 0.10 m and is released on a frictionless surface, then slides up a frictionless ramp. Find the launch speed and the height reached.
Show the solution
$\tfrac12 kx^2 = \tfrac12 mv^2 \Rightarrow v = x\sqrt{k/m} = 0.1\sqrt{1600} = 4.0$ m/s. Height: $mgh = \tfrac12 kx^2 = 4.0$ J $\Rightarrow h = 4/(0.5\cdot9.8) = 0.82$ m.
Common mistakes
- Using energy conservation when friction acts and forgetting the −fₖd term.
- Mixing the zero level for U mid-problem.
- Using x for the spring stretch measured from the wrong point (it is from the natural length).
Exam tipWrite the energy bar chart: K and U at the start, K and U at the end, plus any friction loss. Then the equation writes itself.
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