Further Maths Statics of rigid bodies Practice Questions

Free Further Maths Statics of rigid bodies practice questions with full step-by-step worked solutions. Covers non-uniform-rod, two-strings, moments, beam. Practise exam-style problems and check your method.

non-uniform-rodtwo-stringsmomentsbeamsupportsladder
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
A non-uniform rod ABAB of weight 4040 N and length 44 m rests horizontally in equilibrium, suspended by two vertical light strings attached at the ends AA and BB. The centre of mass of the rod is a distance 1.51.5 m from AA. Find the tension in the string at AA.
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Worked solution

  1. Take moments about AA

    TB×4=40×1.5T_B\times 4=40\times 1.5

    Taking moments about AA removes TAT_A from the equation.

  2. Solve for the tension at BB

    TB=15 NT_B=15\ \text{N}

    Divide the total moment about AA by the length of the rod.

  3. State the answer

    TA=25T_A=25

    This is the required tension, in newtons.

Answer
TA=25 NT_A=25\ \text{N}
Question 2
2 markseasy
A uniform solid cuboid of width 22 m and height 55 m stands on a rough plane. The coefficient of friction between the cuboid and the plane is 0.250.25. The inclination of the plane to the horizontal is slowly increased from zero. Assuming it does not slide, find the angle of inclination at which the cuboid is on the point of toppling. Give your answer to 33 significant figures.
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Worked solution

  1. Locate the centre of mass

    centre of mass at height 125, offset 122\text{centre of mass at height }\tfrac{1}{2}5,\ \text{offset }\tfrac{1}{2}2

    A uniform cuboid has its centre of mass at its centre.

  2. Find when the weight line reaches the lower edge

    tan⁡β=122125=25\tan\beta=\frac{\tfrac{1}{2}2}{\tfrac{1}{2}5}=\frac{2}{5}

    The cuboid topples when the vertical through the centre of mass passes the edge.

  3. Evaluate the toppling angle

    tan⁡β=0.4\tan\beta=0.4

    The ratio of half-width to half-height gives the tangent of the toppling angle.

  4. State the answer

    β=21.8∘\beta=21.8^{\circ}

    This is the required critical value.

Answer
β=21.8∘\beta=21.8^{\circ}
Question 3
4 marksintermediate
A uniform solid cuboid of width 33 m and height 44 m stands on a rough plane, the coefficient of friction being 0.50.5. The inclination of the plane is slowly increased from zero. Determine whether the cuboid slides or topples first.
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Worked solution

  1. Write the critical angle for sliding

    tan⁡λ=μ=0.5\tan\lambda=\mu=0.5

    Sliding starts when the inclination reaches arctan⁡μ\arctan\mu.

  2. Write the critical angle for toppling

    tan⁡β=34=0.75\tan\beta=\frac{3}{4}=0.75

    Toppling starts when the inclination reaches arctan⁡ab\arctan\frac{a}{b}.

  3. Compare the two critical tangents

    μ<34\mu<\frac{3}{4}

    The smaller critical angle is reached first.

  4. State the conditions for the equilibrium of a rigid body

    ΣFx=0,ΣFy=0,ΣM=0\Sigma F_x=0,\quad\Sigma F_y=0,\quad\Sigma M=0

    A rigid body is in equilibrium when the forces and the moments both balance.

  5. Recall the definition of the moment of a force

    M=FdM=Fd

    The moment is the force multiplied by the perpendicular distance to the pivot.

  6. Recall the law of friction at a rough contact

    F≤μRF\le\mu R

    The friction can take any value up to μR\mu R; at the point of slipping it equals μR\mu R.

  7. State the conclusion

    sliding first\text{sliding first}

    The block slides before it topples.

Answer
slides first\text{slides first}
Question 4
6 markshard
A uniform solid cuboid of width 55 m and height 66 m stands on a rough plane, the coefficient of friction being 0.30.3. The inclination of the plane is slowly increased from zero. Determine whether the cuboid slides or topples first.
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Worked solution

  1. Write the critical angle for sliding

    tan⁡λ=μ=0.3\tan\lambda=\mu=0.3

    Sliding starts when the inclination reaches arctan⁡μ\arctan\mu.

  2. Write the critical angle for toppling

    tan⁡β=56=0.8333\tan\beta=\frac{5}{6}=0.8333

    Toppling starts when the inclination reaches arctan⁡ab\arctan\frac{a}{b}.

  3. Compare the two critical tangents

    μ<56\mu<\frac{5}{6}

    The smaller critical angle is reached first.

  4. State the conditions for the equilibrium of a rigid body

    ΣFx=0,ΣFy=0,ΣM=0\Sigma F_x=0,\quad\Sigma F_y=0,\quad\Sigma M=0

    A rigid body is in equilibrium when the forces and the moments both balance.

  5. Recall the definition of the moment of a force

    M=FdM=Fd

    The moment is the force multiplied by the perpendicular distance to the pivot.

  6. Recall the law of friction at a rough contact

    F≤μRF\le\mu R

    The friction can take any value up to μR\mu R; at the point of slipping it equals μR\mu R.

  7. Note that the weight of a uniform body acts at its centre

    W acts at the midpointW\text{ acts at the midpoint}

    A uniform rod or lamina has its centre of mass at its geometric centre.

  8. Recall that a smooth contact has no friction

    F=0 at a smooth contactF=0\text{ at a smooth contact}

    A smooth surface can only push at right angles to itself.

  9. Choose a pivot that removes an unknown force

    take moments about the line of an unknown\text{take moments about the line of an unknown}

    Taking moments through an unknown makes that force contribute no moment.

  10. Resolve the forces into horizontal and vertical components

    components: Fcos⁡θ, Fsin⁡θ\text{components: }F\cos\theta,\ F\sin\theta

    Each force is split along two perpendicular directions before balancing.

  11. Recall that the perpendicular distance uses the sine of the angle

    d=ℓsin⁡θd=\ell\sin\theta

    The moment arm of a vertical force about the foot of an inclined rod is the horizontal offset.

  12. Recall the tangent of an angle from a right triangle

    tan⁡θ=oppositeadjacent\tan\theta=\frac{\text{opposite}}{\text{adjacent}}

    A Pythagorean triple fixes the sine and cosine exactly.

  13. State the conclusion

    sliding first\text{sliding first}

    The block slides before it topples.

Answer
slides first\text{slides first}
Question 5
9 markschallenging
A uniform solid cuboid of width 88 m and height 99 m stands on a rough plane, the coefficient of friction being 0.90.9. The inclination of the plane is slowly increased from zero. Determine whether the cuboid slides or topples first.
Show worked solution

Worked solution

  1. Write the critical angle for sliding

    tan⁡λ=μ=0.9\tan\lambda=\mu=0.9

    Sliding starts when the inclination reaches arctan⁡μ\arctan\mu.

  2. Write the critical angle for toppling

    tan⁡β=89=0.8889\tan\beta=\frac{8}{9}=0.8889

    Toppling starts when the inclination reaches arctan⁡ab\arctan\frac{a}{b}.

  3. Compare the two critical tangents

    μ>89\mu>\frac{8}{9}

    The smaller critical angle is reached first.

  4. State the conditions for the equilibrium of a rigid body

    ΣFx=0,ΣFy=0,ΣM=0\Sigma F_x=0,\quad\Sigma F_y=0,\quad\Sigma M=0

    A rigid body is in equilibrium when the forces and the moments both balance.

  5. Recall the definition of the moment of a force

    M=FdM=Fd

    The moment is the force multiplied by the perpendicular distance to the pivot.

  6. Recall the law of friction at a rough contact

    F≤μRF\le\mu R

    The friction can take any value up to μR\mu R; at the point of slipping it equals μR\mu R.

  7. Note that the weight of a uniform body acts at its centre

    W acts at the midpointW\text{ acts at the midpoint}

    A uniform rod or lamina has its centre of mass at its geometric centre.

  8. Recall that a smooth contact has no friction

    F=0 at a smooth contactF=0\text{ at a smooth contact}

    A smooth surface can only push at right angles to itself.

  9. Choose a pivot that removes an unknown force

    take moments about the line of an unknown\text{take moments about the line of an unknown}

    Taking moments through an unknown makes that force contribute no moment.

  10. Resolve the forces into horizontal and vertical components

    components: Fcos⁡θ, Fsin⁡θ\text{components: }F\cos\theta,\ F\sin\theta

    Each force is split along two perpendicular directions before balancing.

  11. Recall that the perpendicular distance uses the sine of the angle

    d=ℓsin⁡θd=\ell\sin\theta

    The moment arm of a vertical force about the foot of an inclined rod is the horizontal offset.

  12. Recall the tangent of an angle from a right triangle

    tan⁡θ=oppositeadjacent\tan\theta=\frac{\text{opposite}}{\text{adjacent}}

    A Pythagorean triple fixes the sine and cosine exactly.

  13. State the modelling assumptions

    rigid body, light string, uniform where stated\text{rigid body, light string, uniform where stated}

    These are the standard assumptions behind every statics calculation.

  14. Note that the reaction at a hinge has two components

    R=H2+V2R=\sqrt{H^{2}+V^{2}}

    A hinge (or pin) can push in any direction, so it has a horizontal and a vertical part.

  15. Recall how to combine perpendicular components into a magnitude

    ∣R∣=H2+V2|\mathbf{R}|=\sqrt{H^{2}+V^{2}}

    The resultant of two perpendicular components is found by Pythagoras.

  16. Recall how to find the direction of a resultant

    tan⁡ϕ=VH\tan\phi=\frac{V}{H}

    The angle to the horizontal is the arctangent of the vertical over the horizontal part.

  17. Check that the three equilibrium equations are consistent

    3 equations, 3 unknowns\text{3 equations, 3 unknowns}

    Coplanar equilibrium gives exactly three independent scalar equations.

  18. State the conclusion

    toppling first\text{toppling first}

    The block topples before it slides.

Answer
topples first\text{topples first}

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