A spring with stiffness k = 400 N/m is first compressed 0.2 m from its natural length, then compressed 0.4 m. How do the stored potential energies compare?
AThe second compression stores twice the energy
BThe second compression stores three times the energy
CThe second compression stores four times the energy
DThe second compression stores the same energy because k is unchanged
Spring potential energy is V_e = ½kx², which is quadratic in deformation x. Doubling x multiplies the energy by 2² = 4: V_e(0.4) = ½(400)(0.16) = 32 J versus V_e(0.2) = ½(400)(0.04) = 8 J. This quadratic relationship means small increases in deformation cause disproportionately large increases in stored energy.
Question 2 True / False
When friction acts on a sliding particle, total mechanical energy T + V is conserved as long as the particle eventually returns to its starting position.
TTrue
FFalse
Answer: False
Friction is non-conservative: it converts mechanical energy to heat regardless of path or final position. Each pass over the surface dissipates energy, so the particle always arrives back with less mechanical energy than it started with. Conservation of energy T + V = constant holds only when all forces are conservative (gravity, ideal springs). With friction the correct form is T₁ + V₁ + U₁₋₂(nc) = T₂ + V₂, where U₁₋₂(nc) is the (negative) work done by friction.
Question 3 Short Answer
A particle slides down a rough incline from rest. You want to find its speed at the bottom using the work-energy principle. What term must you include that you could omit if the surface were frictionless?
Think about your answer, then reveal below.
Model answer: The work done by friction, U(nc) = −f·d (negative, since friction opposes motion), must be added to the left side of the energy equation. Without it the calculation overcounts the energy available to become kinetic energy.
On a frictionless incline, T₁ + V₁ = T₂ + V₂ suffices because all forces are conservative. Friction does negative work equal to the friction force times the distance traveled along the slope, reducing the final kinetic energy. Omitting this term would predict a speed higher than actually occurs.