Short on time? This page is the recap. For the graphs and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Measure displacement x from equilibrium. The largest displacement is the amplitude A.
- Released from rest at x = +A: use x = A cos(2πft). Passing through equilibrium moving +x: use x = A sin(2πft).
- The angle 2πft is in radians.
- Velocity is the slope of the x–t graph: zero at peaks and troughs, steepest at x = 0.
- Acceleration comes from Newton’s second law with the restoring force: it is opposite to x and proportional to it.
- Amplitude does not change the period. Twice the amplitude means twice the force, twice the acceleration and twice the top speed, so the longer path takes the same time.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| x = A cos(2πft) | x, A in m; f in Hz; t in s | position when released from +A at t = 0 |
| x = A sin(2πft) | as above | position when passing x = 0 in +x at t = 0 |
| T = 1/f | s, Hz | switch between period and frequency |
| a_x = −(k/m)x | m/s² | acceleration of a spring–object system at any x |
| a_max = kA/m | m/s², at x = ±A | largest acceleration |
| slope of a–x graph = −k/m | s⁻² | test for SHM and find k/m |
Where things are zero or largest
| Position | x | Speed | a_x |
|---|---|---|---|
| x = +A | max | 0 | most negative |
| x = 0 | 0 | max | 0 |
| x = −A | min | 0 | most positive |
Assumptions behind the numbers
- The restoring force is proportional to displacement (ideal spring, or a pendulum at a small angle).
- No friction or air resistance, so the amplitude stays constant.
- x is measured from equilibrium. For a hanging spring, that is where the object hangs at rest, not the spring’s unstretched end.
Mistakes to avoid
- Calculator in degrees.
- Saying a = 0 at the turning point because v = 0.
- Putting the largest acceleration at equilibrium.
- Reading T from peak to trough (that is T/2).
- Assuming a bigger amplitude gives a longer period.
- Using cos when the motion starts at equilibrium.
Quick self-check
- x = (0.050 m) cos(2π × 2.0 Hz × t). Where is the object at t = 0.125 s? (At x = 0: the angle is π/2, a quarter cycle.)
- A 0.50 kg object on a 20 N/m spring is at x = +0.030 m. What is a_x? (−1.2 m/s²)
- The amplitude of a spring–object system is tripled. What happens to the period? (Nothing: it stays the same.)
Next: practice questions.