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Flashcards · AP® Physics C: Mechanics · Unit 7

Unit 7: Oscillations flashcards

40 cards on topics 7.1 to 7.5. Pick a topic or study the whole unit.

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7.1 Defining Simple Harmonic Motion (SHM)

What is simple harmonic motion?
Periodic motion in which the net restoring force is proportional to the displacement from equilibrium: F_net,x = −kx with x measured from equilibrium and k > 0 constant.
What differential equation describes SHM?
d²x/dt² = −(k/m)x, from Newton's second law with F_net = −kx. Any variable obeying d²q/dt² = −(constant)q is in SHM.
What is a restoring force?
A force pointing opposite to the displacement, back toward equilibrium. F = +kx points away and is not restoring: that equilibrium is unstable.
Does gravity change SHM for a mass on a vertical spring?
No. A constant force only shifts the equilibrium, to a stretch d = mg/k. Measured from the new equilibrium, the net force is −ky with the same k.
How do you find the effective spring constant from a potential energy curve?
Near a stable equilibrium (a minimum of U), small oscillations are approximately SHM with k_eff = d²U/dx² at the minimum. At a maximum U″ is negative and there is no oscillation.
Why is a simple pendulum only approximately SHM?
Its equation is d²θ/dt² = −(g/ℓ) sin θ. Only for small angles in radians does sin θ ≈ θ, giving the SHM form −(g/ℓ)θ.
Is all periodic motion SHM?
No. Periodic motion repeats in equal time intervals; SHM is the special case with a restoring force proportional to displacement. Test the force law.
Where should displacement be measured from in SHM?
From the equilibrium position, where the net force is zero, not from the relaxed spring length when a constant force also acts.

7.2 Frequency and Period of SHM

How are period, frequency and angular frequency related?
f = 1/T, ω = 2πf = 2π/T. Units: T in s, f in Hz, ω in rad/s.
How do you find the period of any SHM system?
Apply Newton's second law (or τ = Iα), reduce it to d²x/dt² = −ω²x, read off ω² as the constant multiplying x, then T = 2π/ω.
What is the period of a mass on an ideal spring?
T = 2π√(m/k), in any orientation. Gravity does not change it. For a vertical spring with static stretch d = mg/k, this equals T = 2π√(d/g).
What is the period of a simple pendulum?
T = 2π√(ℓ/g) for small angles in radians. The bob's mass cancels, and the formula fails at large angles.
Does the period of SHM depend on amplitude?
No, while the SHM model holds. Larger amplitude gives more energy and higher maximum speed, but the same period.
How do you find k or g from a straight-line graph of period data?
Plot T² against m for a spring: slope 4π²/k. Plot T² against ℓ for a pendulum: slope 4π²/g. Use the slope, not a single reading, especially if there is an intercept.
How does T change if a spring oscillator's mass is multiplied by 9?
T ∝ √m, so T is multiplied by √9 = 3.
How should you time oscillations accurately in a lab?
Time many full cycles and divide by the number of cycles. Count complete cycles, not swings past the middle.

7.3 Representing and Analyzing SHM

What is the position function for SHM?
x = A cos(ωt + φ₀), with ω = √(k/m) from the system and amplitude A and phase φ₀ from the initial conditions. A sine form is equivalent with a phase shift of π/2.
What are velocity and acceleration in SHM?
v_x = −Aω sin(ωt + φ₀) and a_x = −Aω² cos(ωt + φ₀) = −ω²x. An a_x–x graph is a straight line through the origin with slope −ω².
Where are the maximum speed and maximum acceleration in SHM?
v_max = Aω at equilibrium, x = 0, where a = 0. a_max = Aω² at the ends, x = ±A, where the object is momentarily at rest.
What is the speed at any position in SHM?
|v_x| = ω√(A² − x²).
How do you find amplitude and phase from initial conditions?
A = √(x₀² + (v_x0/ω)²), with cos φ₀ = x₀/A and sin φ₀ = −v_x0/(Aω). Use both signs to find the quadrant; tan alone is not enough.
What is resonance?
When a sinusoidal driving force acts at the natural frequency, it adds energy every cycle, so the amplitude grows.
What is v_max for SHM in terms of frequency f?
v_max = Aω = 2πfA, not Af.
An a_x–x graph has slope −49 s⁻². What is the period?
ω² = 49 s⁻², so ω = 7.0 rad/s and T = 2π/7.0 ≈ 0.90 s.

7.4 Energy of Simple Harmonic Oscillators

What is the total energy of a spring–mass oscillator?
E = ½kA² = ½mv_max², constant with no friction. At a turning point it is all potential; at equilibrium it is all kinetic. This gives v_max = A√(k/m) = Aω.
What are K and U at any position in SHM?
U = ½kx² and K = ½k(A² − x²), with x from equilibrium. They always add to ½kA².
Where does K equal U in SHM?
At x = ±A/√2. At x = A/2, U = E/4 and K = 3E/4.
How does oscillator energy depend on amplitude?
E ∝ A². Doubling the amplitude quadruples the energy without changing the period.
How often do K and U repeat in SHM?
Every T/2. For x = A cos ωt, U = ½E[1 + cos 2ωt] and K = ½E[1 − cos 2ωt].
Does a larger mass mean more energy at the same amplitude and spring?
No. E = ½kA² is the same; the larger mass just has a smaller v_max.
Why must the system include the spring when analysing oscillator energy?
The spring stores the potential energy. With the block alone as the system, the spring does work on it and the block's energy changes.
How do you show that SHM conserves energy using calculus?
dE/dt = d(½mv_x² + ½kx²)/dt = v_x(m dv_x/dt + kx) = 0, because Newton's second law gives m dv_x/dt = −kx.

7.5 Simple and Physical Pendulums

What is a physical pendulum?
A rigid body swinging about a fixed axis, with d the distance from the axis to its center of mass. Gravity's restoring torque is −mgd sin θ ≈ −mgdθ for small angles in radians.
What is the period of a physical pendulum?
T = 2π√(I/(mgd)), with ω = √(mgd/I), where I is about the pivot: I = I_cm + md².
How is the simple pendulum a special case of the physical pendulum?
For a point mass on a light string, I = ml² and d = l, so T = 2π√(ml²/(mgl)) = 2π√(l/g).
What is a torsion pendulum?
A body on a twisted wire with restoring torque τ = −κθ, κ in N·m/rad. T = 2π√(I/κ). Gravity plays no part.
What is the period of a uniform rod swinging from one end?
I = mL²/3 and d = L/2, so T = 2π√(2L/(3g)). For L = 1.5 m, T ≈ 2.0 s.
What is the greatest angular speed of a pendulum in SHM?
(dθ/dt)max = ωθ_max at the bottom of the swing, where ω = 2π/T is the fixed angular frequency, not the varying angular velocity.
What is the equivalent length of a physical pendulum?
l_eq = I/(md): a simple pendulum of this length has the same period.
How small must the amplitude be for the small-angle formula?
Below about 15°, the period error is under half a percent. Larger swings have slightly longer periods.

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