A mass on a spring oscillates with amplitude A. The amplitude is then doubled to 2A with the same spring and mass. What happens to the total mechanical energy?
AIt doubles, since energy is proportional to amplitude
BIt stays the same, since energy depends only on the spring constant and mass
CIt quadruples, since total energy E = ½kA² is proportional to A²
DIt halves, since the energy is spread over a longer oscillation path
Total energy in SHM is E = ½kA², so it scales as the square of amplitude. Doubling the amplitude multiplies energy by 2² = 4. This squared relationship is important: a seemingly small increase in amplitude represents a large increase in energy. Option A (doubling) is the common error from assuming a linear relationship. Options B and D are simply wrong — energy is not conserved when you externally change the amplitude, and path length is irrelevant.
Question 2 Multiple Choice
At what position in its oscillation does a spring-mass system have maximum kinetic energy?
AAt the turning points (x = ±A), where the spring stores maximum elastic potential energy
BAt the equilibrium position (x = 0), where the spring is relaxed and all energy is kinetic
CHalfway between the turning point and equilibrium, where energy is equally divided
DAt the same position as maximum potential energy, since they peak together
At x = 0 (equilibrium), the spring has zero deformation, so PE = ½kx² = 0. All the energy must be kinetic: KE = E = ½kA². This is also where the mass moves fastest (v_max = Aω). At the turning points (x = ±A), v = 0, so KE = 0 and all energy is potential. Kinetic and potential energy are perfectly out of phase — they trade off completely, never peaking at the same time. Option D is directly wrong.
Question 3 True / False
The kinetic energy KE(t) and potential energy PE(t) in simple harmonic motion each oscillate at twice the frequency of the displacement x(t).
TTrue
FFalse
Answer: True
Since x(t) = A cos(ωt), we get PE = ½kA²cos²(ωt) and KE = ½kA²sin²(ωt). Both cos²(ωt) and sin²(ωt) oscillate at frequency 2ω — squaring doubles the frequency. This means energy reaches its maximum (kinetic at equilibrium, potential at turning points) twice per displacement cycle. Displacement has one amplitude maximum per cycle; energy has two. This factor-of-two relationship is a precise mathematical consequence of the squared relationship between energy and displacement.
Question 4 True / False
In simple harmonic motion, the total mechanical energy is maximum at the turning points (x = ±A) and minimum at equilibrium (x = 0).
TTrue
FFalse
Answer: False
Total mechanical energy in SHM is CONSTANT — it neither increases nor decreases. E = ½kA² at every point in the motion. What changes between the turning points and equilibrium is the distribution of energy between kinetic and potential forms, not the total. At turning points, all energy is potential; at equilibrium, all energy is kinetic; everywhere else, the sum PE + KE = E is the same constant value.
Question 5 Short Answer
A damped oscillator's amplitude decreases by 30% due to friction (from A to 0.70A). By what fraction does the total energy decrease, and why?
Think about your answer, then reveal below.
Model answer: The total energy decreases to (0.70)² = 0.49 of its original value — a reduction of about 51%. Since E = ½kA², energy is proportional to the square of amplitude. A 30% decrease in amplitude produces a 51% decrease in energy, not a 30% decrease. This squared relationship means energy always falls faster than amplitude when damping is present.
This result has a practical consequence: when monitoring a damped system, a small decrease in the observable amplitude corresponds to a much larger fractional loss of energy. Engineers and physicists use this relationship to infer energy dissipation rates from amplitude measurements. It's also why amplitude is sometimes called 'the energy parameter' — it encodes the system's energy through this precise squared relationship, so knowing one immediately tells you the other.