WME03 Mechanics 3

Lessons in WME03

1. Acceleration as a function of time
   Integrate a to get v and v to get x, using the starting values to find each constant. For distance, split the interval where v = 0.
2. Acceleration as a function of displacement
   When a is a function of x, use a = v dv/dx and integrate to get ½v² = ∫a dx + c.
3. Velocity as a function of position or time
   dx/dt = f(t): integrate. dx/dt = f(x): separate, ∫dx/f(x) = ∫dt, then use the starting values.
4. Hooke's law for elastic strings and springs
   T = λx/l, with x the extension (or compression). Strings go slack; springs can push.
5. Equilibrium problems with elastic strings
   Write each tension with Hooke's law using the unknown length, then resolve or take moments.
6. Elastic potential energy
   EPE = λx²/(2l). Work done stretching further is the difference of two EPEs.
7. Work-energy problems with elastic strings
   List every energy at the start and end, subtract work done against resistance, and set them equal.
8. Newton's second law with a variable force
   F(t): m dv/dt = F. F(x): m v dv/dx = F. F(v): separate m dv/dt = F(v).
9. Newton's law of gravitation
   F = mgR²/x². With v dv/dx = -gR²/x² you find speeds and greatest heights; escape needs U² ≥ 2gR.
10. Simple harmonic motion: definition and proof
    SHM: ẍ = -ω^2x about a centre. Prove it at a general point and conclude with "so SHM". T = 2π/ω.
11. SHM formulae: speed, acceleration and amplitude
    Greatest speed aω at the centre; greatest acceleration aω² at the ends; v² = ω²(a² - x²) everywhere.
12. Time and position in SHM
    Start at the centre: x = a sin ωt. Start at an end: x = a cos ωt. Use quarter periods and symmetry.
13. Oscillations on a horizontal string or spring
    Spring: ω² = λ/(ml). Strings: SHM only while taut, then constant speed. Always write tensions at a general position.
14. Vertical oscillations on a string or spring
    Measure x from the equilibrium position: ω² = λ/(ml) = g/e. If the amplitude exceeds e, a string goes slack and gravity takes over.
15. Angular speed and radial acceleration
    v = rω; acceleration towards the centre rω² = v²/r; the resultant of the real forces supplies m rω².
16. The conical pendulum and horizontal circles
    Vertically T cos θ = mg; horizontally T sin θ = m(l sin θ)ω², so T = mlω². Use the radius, not the string length.
17. Motion on banked surfaces
    Resolve vertically (no acceleration) and horizontally (towards the centre). Above the ideal speed friction acts down the slope; below it, up the slope.
18. Motion in a vertical circle
    Energy gives v²; resolving towards the centre gives T or R. At the top T + mg = mv²/a; at the bottom T - mg = mv²/a.
19. Completing the circle and leaving a surface
    String or inside surface: u² ≥ 5ag. Rod or bead: u² > 4ag. A particle leaves a surface, or a string goes slack, where R or T becomes zero.
20. Centres of mass of standard bodies
    Use symmetry to find the axis, then a standard distance, measured from the right point.
21. Centre of mass of a lamina by integration
    x̄ = ∫xy dx/∫y dx; ȳ = ∫½y² dx/∫y dx.
22. Centre of mass of a solid of revolution
    x̄ = ∫y² x dx / ∫y² dx for a solid of revolution about the x-axis.
23. Centres of mass of composite bodies
    Mass ∝ volume; table of parts; removed parts are negative; Σmx = (Σm)x̄.
24. Bodies suspended in equilibrium
    A suspended body hangs with its centre of mass vertically below the point of suspension; use a right-angled triangle.
25. Rigid bodies on planes: toppling and sliding
    Topple: vertical through G passes outside the base (tan α > d/h). Slide: tan α > μ. The smaller angle wins.

Next exams for WME03

• January 2027: Mon 25 Jan 2027, afternoon, 1h 30m
• Summer 2027: Thu 10 Jun 2027, afternoon, 1h 30m

From Pearson's final timetables. All exam dates. Always confirm your entries with your exams officer.