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A-Level Further Maths: Further Mechanics Practice

A-Level Further Maths — Further Mechanics Practice

16 MCQ practice problems. Select an answer, submit, and review the explanation.

What These Questions Test

These problems test your ability to apply advanced mechanics concepts to quantitative scenarios. You will solve differential equations for oscillations, analyse orbital mechanics, and work with variable forces.

Typical question types:

  • SHM: Given a=−ω2xa = -\omega^{2}x, find the period, amplitude, and maximum speed. Relate displacement, velocity, and acceleration. Determine the energy at different positions.
  • Damped oscillations: Identify the damping regime (underdamped, critically damped, overdamped) from the parameters. Find the decay rate and damped frequency. Sketch displacement-time graphs.
  • Resonance: Determine the amplitude at resonance for a given driving force. Describe how damping affects the resonance curve. Relate Q-factor to the sharpness of the resonance peak.
  • Gravitation: Calculate gravitational field strength, potential, or escape velocity at a given distance from a planet. Use energy conservation to find orbital speeds. Apply Kepler’s third law to compare orbits.
  • Variable forces: Use F=mvdvdxF = mv\frac{dv}{dx} to find velocity as a function of position. Calculate work done by integrating force over distance. Apply energy methods when force is not constant.

Approach Strategy

  1. Write the differential equation. For oscillation problems, always start by writing mx¨+cx˙+kx=F(t)m\ddot{x} + c\dot{x} + kx = F(t) and identifying the coefficients.
  2. Check the units. Gravitational calculations involve very large numbers (G=6.67×10−11G = 6.67 \times 10^{-11}). Ensure you are using SI units throughout.
  3. Use energy methods when possible. For variable forces, energy conservation (12mv2+V(x)=const\frac{1}{2}mv^{2} + V(x) = \text{const}) is often simpler than integrating force directly.
  4. Verify limiting cases. If your answer for orbital speed does not decrease with increasing radius, something is wrong.

Intuition

For oscillations, think of the differential equation as a recipe. The mass (mm), damping (cc), and spring constant (kk) determine the ingredients. The solution tells you how the system evolves. Damping is like stirring — it gradually removes energy. Resonance is like pushing a swing at the right frequency to make it go higher.

For gravitation, remember that gravity is always attractive. The potential is negative because you must add energy to move a mass away to infinity (where V=0V = 0). The deeper you are in a gravitational well (more negative VV), the harder it is to escape.

Common Mistakes

  1. Using the natural frequency when the damped frequency is required. For a damped system, the oscillation frequency is ωd=ω02−γ2\omega_{d} = \sqrt{\omega_{0}^{2} - \gamma^{2}}, which is less than the natural frequency ω0\omega_{0}.
  2. Forgetting the negative sign in gravitational potential. V=−GMrV = -\frac{GM}{r}, not +GMr+\frac{GM}{r}. This affects all energy calculations including escape velocity.
  3. Applying F=maF = ma when force is not constant. For variable forces, you must use F=mdvdtF = m\frac{dv}{dt} or F=mvdvdxF = mv\frac{dv}{dx} and integrate. Using F=maF = ma with the initial force gives the wrong answer.
  4. Confusing orbital speed with escape speed. Orbital speed: v=GMrv = \sqrt{\frac{GM}{r}}. Escape speed: ve=2GMr=v2v_{e} = \sqrt{\frac{2GM}{r}} = v\sqrt{2}. The escape speed is 2\sqrt{2} times the orbital speed.

Cross-References