Add the oscillations as arrows (phasors). For your sliders the amplitude is A = units and the phase is α = °. The book gets 7 units in both parts: (a) exactly 7, (b) 6.985, rounded to 7.
Problem statement
Add oscillations of the same direction and the same frequency. (a) x1 = 3 cos(ωt + π/3) and x2 = 8 sin(ωt + π/6). (b) x1 = 3 cos ωt, x2 = 5 cos(ωt + π/4) and x3 = 6 sin ωt. Find the amplitude of the resultant oscillation in each case.
We assume all oscillations have the same ω. This means the sum is again a single harmonic oscillation at that ω.
We assume x1 in part (a) is 3 cos(ωt + π/3). The printed sign is garbled in the source. This reading gives exactly the book answer of 7.
We assume a sine is a cosine shifted by 90°: sin(ωt + θ) = cos(ωt + θ − 90°). This means sin(ωt + π/6) has phase −60°, and sin ωt has phase −90°.
Symbols: A_i and φ_i are the amplitude and phase of oscillation i. X and Y are the sums of A cos φ and A sin φ. A and α are the amplitude and phase of the sum.
The big idea in plain words
Picture each oscillation as an arrow of length A turning at rate ω. The oscillation x is the shadow of the arrow on a line. All arrows turn together, so the picture never changes shape. The sum of the shadows is the shadow of the sum of the arrows. So add the arrows head to tail, and the length of the long arrow is the new amplitude.
The solution in each frame
Lab frame
The point oscillates along one line fixed to the lab. All the work is in this frame, in the next section.
Other frames
No second frame is needed. The problem is only about adding functions of time, with no moving observer.
The solution step by step
Write every term as A cos(ωt + φ). Then each term has a length A and an angle φ. Use the sum rule cos(ωt + φ) = cos ωt cos φ − sin ωt sin φ for each term and add them.
A single oscillation A cos(ωt + α) expands the same way, to A cos α cos ωt − A sin α sin ωt. Matching the two forms gives
Squaring and adding X and Y, and using cos(φi − φj) = cos φi cos φj + sin φi sin φj, gives the cosine rule for any number of terms:
Part (a). The phases are +60° and −60°, so the gap is 120°. A1 = 3 and A2 = 8.
Part (b). The phases are 0°, 45° and −90°. The components are
Your sliders now give X = , Y = , A = and α = °. The sum of the squares and cross terms is , which equals A².
Book: (a) A = 7, (b) A ≈ 7. The page gives exactly 7 for (a) and 6.985 for (b), which rounds to 7. The printed x1 in (a) has a garbled sign, and the reading 3 cos(ωt + π/3) is the one that matches the book. At the time shown, x1 + x2 + x3 = and A cos(ωt + α) = . A finite-difference slope of x gives against the formula .
Five methods
1. Force method
This does not suit the problem. No force is given, and the question is only about adding functions of time.
2. Energy method
The energy of an oscillator is proportional to A². So A² for the sum is the sum of the separate A² values plus cross terms that depend on the phase gaps. In phase the cross terms add energy. In opposite phase they take it away.
3. Graph method
Draw the arrows head to tail, each at its angle φ. The arrow from the start to the end is the resultant. Its length is A and its angle is α. The page draws this at t = 0, then turns the whole chain at ω.
4. Calculus method
Expand each cosine with the sum rule, collect the cos ωt and sin ωt parts, and match them to A cos(ωt + α). This gives X, Y, A and α exactly, for any number of terms.
5. Check method
All terms have the same units, so A has the same units. In phase (all φ equal) the result is ΣA. In opposite phase with two terms it is |A1 − A2|. At a quarter turn apart it is √(A1² + A2²). Three equal arrows at 120° cancel to 0. The page has a preset for each.
Fastest: the cosine rule for two terms, the components for three. Most reliable: the components method, because it works for any number of terms.
What each slider changes
| Control | Changes the answer? | What it changes |
|---|---|---|
| A1, A2, A3 | Yes | The arrows, X, Y, A, α and every graph |
| φ1, φ2, φ3 | Yes | The arrow directions, A, α, the shapes of the curves. Only differences of phase change A. |
| Period T | Not A | ω, the time axis, speeds and the motion speed |
| Screen speed | No | Only how fast the animation runs |
| Target amplitude (given-answer mode) | Sets the answer | Solves the phase gap φ2 − φ1 for two oscillations |
| Zoom, Auto-fit | No | The size of the arrows in the picture only |
Parameters
| Quantity | Symbol | Value used |
|---|---|---|
| Amplitudes | A1, A2, A3 | , , |
| Phases | φ1, φ2, φ3 | °, °, ° |
| Period | T | s |
| Angular frequency | ω | rad/s |
| Resultant amplitude and phase | A, α | , ° |
| Largest and smallest possible A | and |
Exam traps
- Adding the amplitudes directly. That is right only when all phases are equal.
- Forgetting that a sine is a shifted cosine. sin ωt has phase −90°, not 0°.
- Mixing degrees and radians in the cosine of the phase gap.
- Taking α = atan(Y/X) without checking the quadrant. Use atan2.
- Rounding too early. In (b) the exact amplitude is 6.985, and the book rounds it to 7.
- Adding oscillations of different ω as if they gave one amplitude. They do not.