Further Mathematics · Pearson Edexcel 9FM0 · Option FM2 · Topic 3

Further centres of mass

Centres of mass of solids, and when objects hang in equilibrium, slide or topple.

  • 3 sub-topics
  • Taught in Year 13
  • 51 practice questions
  • 3 animated lessons
Practice questions (51) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 3.1Centres of mass by integrationY13
  2. 3.2Equilibrium of rigid bodiesY13
  3. 3.3Toppling and slidingY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 51 questions in this topic, free, no account needed.

Further centres of mass

Not needed for Year 12Easy

A freely suspended lamina · Year 13 · 1 mark

  1. A lamina is freely suspended from a point and hangs at rest. Where is its centre of mass?
    (1)

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  1. The weight and the force at the pivot are then in the same vertical line, so there is no turning effect.A: Vertically below the point of suspension

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Further centres of mass

Not needed for Year 12Medium

Toppling and sliding · Year 13 · 5 marks

A uniform rectangular block, of width 5 cm and height 6 cm, rests with a 5 cm edge on a rough plane. The coefficient of friction is 0.5. The angle of the plane, α\alpha, is gradually increased.
  1. (a)
    Find the value of tan⁡α\tan \alpha at which the block would slide, if it did not topple first.
    (2)
  2. (b)
    Find the value of tan⁡α\tan \alpha at which the block would topple, if it did not slide first.
    (2)
  3. (c)
    Hence state what happens first, and at what angle.
    (1)

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Part (a)2 marks

  1. μR\mu R=mgsin⁡α, R{}= mg\sin\alpha, \ R=mgcos⁡α{}= mg\cos\alpha
  2. tan⁡α=0.5\tan\alpha = 0.5

Part (b)2 marks

  1. It topples when the centre of mass is vertically above the lower edge.
  2. tan⁡α\tan\alpha=5/26/2{}= \frac{5/2}{6/2}=56{}= \frac{5}{6}

Part (c)1 mark

  1. The smaller value of tan⁡α\tan \alpha is reached first.It slides, at α=26.6∘\alpha = 26.6^\circ.

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Further centres of mass

Not needed for Year 12Hard

Centre of mass by integration · Year 13 · 6 marks

The region RR is bounded by the curve y=x3y = x^{3}, the xx-axis and the line x=1x = 1. A uniform lamina has the shape of RR.
  1. Find the exact coordinates of the centre of mass of the lamina.
    (6)

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  1. area\text{area}=∫01x3 dx{}= \int_0^{1} x^{3}\,\mathrm{d}x=14{}= \frac{1}{4}
  2. xˉ\bar{x}=1A∫01xy dx{}= \frac{1}{A}\int_0^{1} x y\,\mathrm{d}x
  3. xˉ=45\bar{x} = \frac{4}{5}
  4. The strip's centre is at half its height.yˉ\bar{y}=1A∫0112y2 dx{}= \frac{1}{A}\int_0^{1} \tfrac{1}{2}y^2\,\mathrm{d}x
  5. yˉ=27\bar{y} = \frac{2}{7}

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