Questions: Simple Harmonic Motion and Natural Frequency

Question 1

A spring-mass system is oscillating with amplitude A. If you increase the initial displacement to 2A without changing the spring or mass, what happens to the natural frequency?

• A. It doubles — the system now oscillates with twice the energy, so it moves faster
• B. It increases by a factor of √2 — energy is proportional to A², so frequency scales accordingly
• C. It decreases — the larger displacement means the mass takes more time to travel the full cycle
• D. It remains unchanged — natural frequency depends only on k and m, not on amplitude

Answer: (D)

Natural frequency ω_n = √(k/m) is a property of the system's physical parameters alone — it is completely independent of amplitude. Doubling the amplitude doubles the maximum velocity and quadruples the total energy, but both changes cancel: the mass travels twice as far at twice the speed, so the period is unchanged. This amplitude-independence is a unique and important feature of systems with linear restoring forces.

Question 2

An engineer uses the energy method to find the natural frequency of a compound pendulum. She writes E = ½Iθ̇² + ½k_eff θ², differentiates with respect to time, and sets dE/dt = 0. What equation does she arrive at?

• A. Iθ̈ + k_eff θ = 0, revealing ω_n = √(k_eff/I)
• B. Iθ̈ = k_eff θ, which requires numerical solution for ω_n
• C. θ̈ = 0, indicating no oscillation occurs
• D. dE/dt = 0 only holds at equilibrium and gives no information about frequency

Answer: (A)

Differentiating E = ½Iθ̇² + ½k_eff θ² gives dE/dt = Iθ̇θ̈ + k_eff θθ̇ = 0. Dividing through by θ̇ (nonzero except at turning points) yields Iθ̈ + k_eff θ = 0 — the standard SHM equation. Comparing to the form θ̈ + ω_n²θ = 0 directly gives ω_n = √(k_eff/I). This energy method bypasses drawing free body diagrams and applying Newton's second law directly, often simplifying the algebra for complex geometries.

Question 3

The natural frequency of a spring-mass system is independent of the amplitude of oscillation.

• T. True
• F. False

Answer: True

ω_n = √(k/m) contains no amplitude term. This holds for any system with a truly linear restoring force (F = −kx), and is the defining characteristic that makes SHM exactly sinusoidal with a constant period. Amplitude determines the energy stored (E = ½kA²) and the maximum speed (v_max = Aω_n), but not how fast the cycle repeats. This property breaks down for large displacements where Hooke's law no longer applies — hence the small-angle approximation for pendulums.

Question 4

Attaching a heavier mass to a spring increases the natural frequency because more mass stores more kinetic energy and therefore oscillates faster.

• T. True
• F. False

Answer: False

A heavier mass decreases the natural frequency: ω_n = √(k/m) — mass m is in the denominator. Greater mass means greater inertia, so the restoring force accelerates it more slowly and the oscillation is slower. The misconception confuses energy storage with oscillation rate. While more mass does carry more kinetic energy at peak velocity, it also moves more sluggishly — the net effect is a lower natural frequency and a longer period.

Question 5

Explain why the energy method (setting dE/dt = 0 for a conservative system) yields the natural frequency, and why it is sometimes preferred over Newton's second law for this purpose.

Model answer: For a conservative system, total mechanical energy E = KE + PE is constant, so dE/dt = 0. Differentiating and collecting terms produces an equation of the form ẍ + ω_n²x = 0, where ω_n² is the coefficient of x (or θ for rotational systems). Reading off this coefficient directly gives ω_n without requiring explicit identification of all forces. The energy method is preferred when the system has complex geometry — pulleys, compound pendulums, systems with multiple moving parts — because energy is scalar and easier to write down correctly than all the force and moment equations.

Newton's second law requires drawing free body diagrams and resolving forces, which can be algebraically messy for complex systems. Energy methods bypass this by working with scalar quantities (kinetic and potential energy), which are often easier to express for complicated geometries. Both methods must give the same ω_n — they are mathematically equivalent for linear, conservative systems. The energy method is especially powerful when the system's kinetic energy involves multiple velocity components or when constraint forces do no work and can simply be ignored.