1.6.3 · D5Oscillations & Waves
Question bank — ω, T, f relationships
Before we start, three plain-word anchors (never assume them):
- Cycle = one complete repeat of the motion — back to the same position moving the same way.
- Radian = a way of measuring angle where one full turn around a circle equals of them (about ). It is a pure number, not a "real" unit like the metre.
- Phase = the angle sitting inside the cosine in — it tells you where in the cycle you are.
Two pictures anchor the whole page. Keep them in view as you work through the traps.
The triangle above shows the three quantities as corners linked by their conversions — the lives only on the two edges touching . The circle picture below shows why: an oscillation is the shadow of a point going round a circle, so one lap ( rad) equals one cycle.
True or false — justify
A doubling of frequency doubles the angular frequency .
True — since is a straight proportionality (a line through the origin with slope ), scaling scales by the same factor; the is a fixed conversion constant that never changes, so it cannot distort the doubling.
A doubling of frequency doubles the period .
False — is inverse, so doubling halves . Physically: if you cram twice as many cycles into each second, each individual cycle must occupy half the time.
and have the same units.
False — is cycles per second (Hz), is radians per second. They only look alike because a radian is dimensionless (a pure ratio), but the physical content "cycles" vs "radians" differs by exactly the radians in one cycle.
Period and frequency are reciprocals of each other.
True — one cycle takes seconds, so in one second you fit cycles, and that count is ; hence . The two are the same fact read in opposite directions (seconds-per-cycle vs cycles-per-second).
A larger angular frequency always means a shorter period.
True — , so bigger (faster angle-sweeping) means the fixed target of radians is swept sooner, giving a smaller . It is a strict inverse, so the statement holds for every positive .
Two oscillators with the same must have the same .
True — depends only on (nothing about amplitude, mass, or phase enters), so equal periods force equal angular frequencies. Amplitude and phase can differ freely and it changes nothing.
Changing the amplitude changes the frequency of an SHM.
False — in ideal SHM never enters . For a spring, (with stiffness and mass ): the restoring force grows in proportion to displacement, so a bigger swing feels a proportionally bigger pull-back and the extra distance is covered in the same time — the rhythm is untouched. See Springs and Pendulums.
The frequency in Hz equals the number of radians swept per second.
False — Hz counts cycles per second; radians swept per second is , which is times larger for the same motion because each cycle is worth radians.
A negative angular frequency is physically meaningless.
False — a sign on just encodes the direction of rotation of the reference circle (clockwise vs anticlockwise). Since , the observed SHM position is identical, so we conventionally take and absorb any direction into the phase ; it is a choice, not a law.
Spot the error
Error: ", since the connects period and frequency."
Wrong pairing — has units of seconds, so it must be the reciprocal . The belongs only with (radians), giving ; look at the triangle figure — the never sits on the edge.
Error: "."
The is in the wrong place: (multiply) or equivalently (divide by period, not frequency). Check units — rad/s (rad) (cycles/s), so it must be a product with .
Error: "In , the frequency is Hz."
is (the coefficient of , in rad/s), not . The frequency is Hz. The number out front is the amplitude , unrelated to rhythm.
Error: "Since is 'angular frequency', I'll set my calculator to degrees when evaluating ."
The phase is in radians, not degrees — degree mode gives a completely wrong SHM curve. Keep the calculator in radian mode for oscillations.
Error: " because both describe how fast the oscillation is."
They both grow with speed but are not equal — counts radians/s and counts cycles/s, and one cycle is radians, so .
Error: "A wave with higher frequency has a longer wavelength at fixed speed."
Backwards — from Wave Speed v = fλ, at fixed the wavelength is inversely related to , so higher frequency means shorter wavelength.
Error: "."
Divide, don't multiply: . You are converting radians/s into cycles/s, so you strip out the radians-per-cycle, not add another.
Error: "A negative frequency means the oscillator has negative energy."
No — a sign flip on (or ) reverses the sense of rotation of the reference circle, not the energy; energy depends on amplitude and , both unaffected by the sign. Physically we keep and put direction into .
Why questions
Why does the factor appear in and nowhere in ?
Because is the "angle per cycle" — it only enters when you translate cycles into radians. and both speak in cycles/seconds, so no angle conversion is needed between them. On the triangle figure, the decorates exactly the two edges meeting at .
Why is angular frequency measured in radians per second rather than degrees per second?
Because SHM comes from projecting Uniform Circular Motion, and radians make the circle's full turn exactly , which is the natural argument for cosine and keeps clean (degrees would force clumsy factors).
Why do we say oscillation is "the shadow of circular motion"?
A point moving steadily around a circle, viewed edge-on, moves back and forth exactly like SHM (see the circle figure — the projection onto the axis traces ); one lap ( rad) equals one oscillation cycle, which is why (a rotation rate) governs a back-and-forth motion. See Uniform Circular Motion.
Why is the quantity that sits inside the cosine, not or ?
Cosine's input must be an angle in radians, and delivers exactly that (rad/s × s = rad). Using or inside would give the wrong argument unless you rebuilt the .
Why can we talk about the same oscillation with three different numbers (, , ) without contradiction?
They describe the same rhythm in different "currencies" — seconds-per-cycle, cycles-per-second, and radians-per-second — linked by fixed conversions (, ), so they must always agree.
Why does the phase advance by exactly over one period?
One period is by definition one full cycle, and one full cycle corresponds to sweeping the whole circle, which is radians — so , matching . See Phase and Phase Difference.
Why does a stiffer spring (larger ) oscillate faster?
Because (stiffness over mass ): a larger spring constant means a stronger restoring pull for the same displacement, so the mass is yanked back sooner, raising and therefore while shrinking . See Springs and Pendulums.
Why does a longer pendulum swing more slowly?
Because (gravity over length ): a bigger length shrinks the ratio , lowering , so the period grows with . The rhythm is set by geometry and gravity, never by the swing's size. See Springs and Pendulums.
Edge cases
What is the period of something that never repeats (e.g. a mass that just sits still)?
There is no finite period — , and blows up to infinity, meaning "one cycle takes forever." A non-oscillating object has zero frequency.
As , what happens to ?
— the angle sweeps so slowly that a full cycle takes unboundedly long, i.e. the motion effectively stops repeating.
As , what happens to and ?
Both go to their limits: (infinitely brief cycles) and (infinitely many cycles per second).
Can the period be zero for a real oscillation?
No — a zero period would need infinite frequency (infinitely fast repeats), which is physically impossible; always for genuine motion.
Is the same physical situation as ?
Yes — since , one is zero exactly when the other is, and both describe "no oscillation" (no cycles, no angle sweeping).
What does flipping the sign of do to the motion ?
Nothing visible — because is an even function, , so the position graph is identical; the sign only records whether the imagined reference point circles clockwise or anticlockwise, which we usually bury inside the phase .
If two SHMs have identical but different phase , do they have the same period?
Yes — the period depends only on (); merely shifts where in the cycle each starts, not how long a cycle lasts. See Phase and Phase Difference.
A pendulum on the Moon swings more slowly than on Earth. Does its increase or decrease?
increases — weaker gravity gives a smaller , and grows as shrinks. See Springs and Pendulums.
Connections
- ω, T, f relationships — the parent note these traps stress-test.
- Simple Harmonic Motion — where lives inside the cosine.
- Uniform Circular Motion — origin of the and of .
- Phase and Phase Difference — the phase probed in several items.
- Wave Speed v = fλ — the frequency–wavelength trap.
- Springs and Pendulums — the amplitude-independence and gravity edge cases.