Topic 5 of 6 · Form 5
Further Mechanics
Equation of trajectory y = x tan θ − gx²/(2u²cos²θ); range on a horizontal plane; maximum heights and two possible angles to hit a target.
Notes for this topic 1 file, in order
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Projectiles
Equation of trajectory y = x tan θ − gx²/(2u²cos²θ); range on a horizontal plane; maximum heights and two possible angles to hit a target.
Equilibrium of rigid bodies
Moments; centres of mass of uniform laminas and solids (hemispheres, cones, composite bodies); toppling and sliding.
Circular motion
Horizontal circles (conical pendulum, banked tracks) and vertical circles (energy conservation, conditions for completing the circle).
Worked example: A particle on a string of length 0.5 m moves in a vertical circle. Find the least speed at the bottom for it to complete the circle (g = 10).
At top v² ≥ gr = 5; energy: u² = v² + 4gr = 5 + 20 = 25 → u = 5 m s⁻¹
Hooke's law and elastic energy
T = λx/l; elastic potential energy λx²/(2l); problems with energy and equilibrium.
Linear motion under variable forces
Using v dv/dx = F/m and dv/dt = F/m.
Momentum and restitution
Newton's experimental law (e) for direct and oblique impacts, including impacts with walls.
Check yourself
- State the condition for a particle on a string to complete a vertical circle.
- An elastic string of natural length 2 m and modulus 40 N is stretched to 2.5 m. Find the tension and energy stored.
- Where is the centre of mass of a uniform solid hemisphere of radius r?
Show answers
- Tension at the top must be ≥ 0, i.e. v² ≥ gr at the top.
- T = 10 N; EPE = 40 × 0.25/4 = 2.5 J
- 3r/8 from the centre of the plane face.
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