Equilibrium, sliding and topplingAQA A-Level Further Maths: Subtopic test

1. 1
   A uniform horizontal rod $AB$ of length $4$ m and mass $6$ kg rests in equilibrium, supported by two vertical strings. One string is attached at $A$ and the other at the point $C$ on the rod, where $AC=3$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string at $C$.

   [1 mark]

   • A$19.6$ N
   • B$29.4$ N
   • C$39.2$ N
   • D$58.8$ N

   (b)

   Find the tension in the string at $A$.

   [1 mark]

   • A$9.8$ N
   • B$19.6$ N
   • C$39.2$ N
   • D$58.8$ N

   (c)

   A particle of mass $3$ kg is now attached to the rod at $B$. Find the new tension in the string at $A$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform rectangular lamina $ABCD$ has $AB=0.9$ m, $AD=0.6$ m and mass $3$ kg. It hangs freely in equilibrium from a smooth pivot at $A$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the angle between $AB$ and the vertical.

   [1 mark]

   • A$33.7^\circ$
   • B$56.3^\circ$
   • C$53.1^\circ$
   • D$36.9^\circ$

   (b)

   A particle of mass $3$ kg is attached at $B$. Find the new angle between $AB$ and the vertical when the lamina hangs freely from $A$.

   [1 mark]

   • A$6.3^\circ$
   • B$24.0^\circ$
   • C$33.7^\circ$
   • D$12.5^\circ$

   (c)

   The lamina is now held in equilibrium with $AB$ horizontal by a vertical string attached at $B$. The lamina can still rotate freely about $A$. Find the tension in the string.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform ladder $AB$ of length $5$ m and mass $20$ kg rests in equilibrium in a vertical plane with its upper end $B$ against a smooth vertical wall and its lower end $A$ on rough horizontal ground. The foot $A$ is $3$ m from the wall. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the force exerted by the wall on the ladder.

   [3 marks]

   (b)

   Find the least coefficient of friction between the ladder and the ground for the ladder to remain in equilibrium.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform solid cuboid block of mass $12$ kg has a rectangular cross-section $0.30$ m wide and $0.80$ m high. The coefficient of friction between the block and any surface it rests on is $0.5$. Take $g=9.8$ m s$^{-2}$.

   (a)

   The block rests with its $0.30$ m edge along a rough plane inclined at $\theta$ to the horizontal, so that its $0.80$ m edges are perpendicular to the plane. The plane is slowly tilted. (i) Show that the block is on the point of toppling when $\tan\theta=\frac38$. (ii) Hence find $\theta$ and decide, with justification, whether the block slides or topples as $\theta$ is increased from $0^\circ$.

   [6 marks]

   (b)

   The block is now placed on a horizontal rough surface. A horizontal force $P$ is applied at the top of one of the $0.80$ m faces, perpendicular to the face, and slowly increased from zero. Find the value of $P$ at which the block topples about its lower edge and the value of $P$ at which it would slide, and hence state which happens first.

   [6 marks]

   Total for question 4: 12 marks

End of questions