DPST1021 Chap.12 Simple Harmonic Motion and Pendulums
Simple Harmonic Motion and Pendulums
Oscillations open the waves block of Physics 1A in week 10, across Lectures 25 and 26. The starting point is a precise definition: a body performs simple harmonic motion when the net force on it is proportional to its displacement from equilibrium and directed back toward it.
From that single condition you obtain a cosine solution for position, and you learn to connect frequency, angular frequency, period, spring constant and mass for a block on a spring, using the relations printed under Waves and Oscillations on the formula sheet.
Energy gives a second way into the same motion.
An undamped oscillator keeps a fixed total equal to half the spring constant times the amplitude squared, and that total is traded back and forth between elastic and kinetic forms twice every cycle. Lecture 26 then projects uniform circular motion onto a diameter to show where the angular frequency comes from, and applies the small-angle approximation to derive periods for simple, physical and torsion pendulums.
Damping and driven oscillations, including resonance, are described in words only.
These ideas feed straight into the next chapters, because every element of a string carrying a sinusoidal wave moves in simple harmonic motion.
They belong to Test 2, which draws its questions from homework quizzes 6 to 10, and Lab 7, Linear Oscillatory Motion, takes the same physics into the laboratory in week 11. Typical questions ask for x(t) from starting conditions or a pendulum period for an extended object.
What this chapter covers
- 01
Definition of SHM: identify a restoring force proportional to displacement and opposite in direction, then write the acceleration as minus omega squared times x.
- 02
Spring oscillator relations: connect omega, frequency and period with the square root of k over m, and see why the period ignores amplitude.
- 03
Energy exchange: track kinetic and elastic energy through a cycle with a constant total of half k A squared, and find speed at any displacement.
- 04
Reference circle: treat SHM as the projection of uniform circular motion and use it to choose the phase constant from initial conditions.
- 05
Pendulum periods: apply small-angle results for simple, physical and torsion pendulums, using the moment of inertia about the pivot for rigid bodies.
- 06
Damping and driving: describe the decaying amplitude of a damped system and the large response near resonance, without calculation.
Worked example · free
Building x(t) from a push and a stretch
- 1Angular frequency: ω = √(k/m) = √(50/2.0) = 5.0 rad/s, so the period is 2π/5.0 = 1.26 s.
- 1Amplitude from both starting values: A = √(x0² + (v0/ω)²) = √(0.0144 + 0.0256) = 0.20 m.
- 1Phase: cos φ = 0.12/0.20 = 0.60, and v0 = −ωA sin φ = −0.80 m/s requires sin φ = +0.80, so φ = 0.93 rad and x(t) = 0.20 cos(5.0t + 0.93) m.
- 1Equilibrium is first reached when 5.0t + 0.927 = π/2, giving t = 0.13 s; there all energy is kinetic and the speed is ωA = 1.0 m/s.
Key terms
- Simple harmonic motion
- Oscillation produced by a restoring force that grows linearly with displacement and points back to equilibrium, giving sinusoidal position against time.
- Angular frequency
- The rate omega at which the phase of an oscillation advances, in radians per second, equal to 2 pi times the frequency.
- Phase constant
- The angle phi inside the cosine at t = 0, fixed by where the oscillator starts and which way it is moving.
- Physical pendulum
- A rigid body swinging about a pivot away from its centre of mass, whose period depends on its moment of inertia.
- Resonance
- A large-amplitude response of a driven oscillator, occurring when the driving frequency is close to the natural frequency.
Simple Harmonic Motion and Pendulums FAQ
Does a larger amplitude make a spring oscillator take longer?
No. The period depends only on mass and spring constant, so pulling the block further simply means it covers more ground at higher speeds in the same time. The independence holds only while the spring obeys Hooke's law; overstretch it past its elastic limit and the result fails.
Why does the mass of a pendulum bob not affect its period?
Gravity's force on the bob and the bob's inertia both scale with its mass, so mass cancels from the equation of motion. Only the string length and g remain. That cancellation fails for a physical pendulum, where the mass distribution matters through the moment of inertia about the pivot and the distance to the centre of mass.
How large can the swing be before the small-angle formula fails?
Accuracy degrades slowly. At an amplitude of 15 degrees, the true period of a simple pendulum is only about 0.4 percent longer than 2 pi root l over g,, smaller than typical timing uncertainty in a teaching lab. Beyond roughly 30 degrees the error reaches a few percent.
Does hanging a spring vertically change its period?
Gravity shifts the equilibrium position down by mg/k, but measured from that new equilibrium the restoring force is still minus k times the displacement. The period is therefore the same as for the horizontal arrangement. A 2.0 kg mass on a 50 N/m spring, for instance, sits 0.39 m lower yet still oscillates with a 1.26 s period.
Exam move
Begin with the definition, because every later result follows from writing the acceleration as minus omega squared times displacement. Practise putting a spring, a swinging bob and a pivoted rod into that form. Next, drill initial conditions on a mix of starts, such as release from rest, a kick from equilibrium and a general push, checking each phase against the sign of the starting velocity.
Keep a one-line energy equation ready for any question that gives a displacement and asks for a speed, since it avoids solving for time. For pendulums, list I about the pivot and d before substituting, then sanity-check against a simple pendulum of similar size. Homework Quiz 8 covers both lectures and allows three tries at each part, so use it to fix significant-figure habits before Test 2.
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