Equilibrium, toppling and slidingEdexcel A-Level Further Maths: Subtopic test

1. 1
   A uniform rectangular lamina $ABCD$ has $AB=12$ cm and $BC=8$ cm. It is freely suspended from the vertex $A$ and hangs in equilibrium.

   (a)

   Which line through $A$ is vertical?

   [1 mark]

   • A$AB$
   • B$AD$
   • C$BD$
   • D$AC$

   (b)

   Find the angle between $AB$ and the vertical.

   [1 mark]

   • A$56.3^\circ$
   • B$18.4^\circ$
   • C$33.7^\circ$
   • D$48.2^\circ$

   (c)

   Find the distance from $A$ to the centre of mass of the lamina.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform solid block has a rectangular cross-section 0.4 m wide and 0.6 m high. It rests on a rough plane inclined at an angle $\alpha$ to the horizontal, with its 0.4 m wide face in contact with the plane and the width lying along a line of greatest slope.

   (a)

   The block is on the point of toppling, with no slipping. Find $\tan\alpha$.

   [1 mark]

   • A$\frac{2}{3}$
   • B$\frac{3}{2}$
   • C$\frac{1}{3}$
   • D$\frac{4}{9}$

   (b)

   The coefficient of friction is 0.5. The angle $\alpha$ is increased slowly from $0^\circ$. What happens?

   [1 mark]

   • AThe block topples when $\alpha=33.7^\circ$
   • BThe block slides when $\alpha=26.6^\circ$
   • CThe block slides when $\alpha=33.7^\circ$
   • DThe block topples when $\alpha=26.6^\circ$

   (c)

   Find the range of values of the coefficient of friction $\mu$ for which the block topples before it slides.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rod $AB$, of mass 12 kg and length 4 m, has its end $A$ on rough horizontal ground and its end $B$ against a smooth vertical wall. The rod lies in a vertical plane perpendicular to the wall and is inclined at $60^\circ$ to the horizontal. Take $g=9.8$ m s⁻².

   (a)

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

   [3 marks]

   (b)

   The rod is in limiting equilibrium. Find the coefficient of friction between the rod and the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform rectangular block of mass 30 kg has a square base of side 0.5 m and a height of 1.2 m. It rests on rough horizontal ground, and the coefficient of friction between the block and the ground is 0.6. A horizontal force of magnitude $P$ newtons is applied to a vertical face, in the vertical plane through the centre of the block and perpendicular to that face. Take $g=9.8$ m s⁻².

   (a)

   The force is applied at a height of 1.0 m above the ground. Find the value of $P$ at which the block is on the point of toppling, and the value at which it is on the point of sliding. State which happens first as $P$ is gradually increased from zero.

   [6 marks]

   (b)

   The force is now applied at a height of $x$ metres above the ground. Find the greatest value of $x$ for which the block slides before it topples, and explain what happens when $x$ is greater than this value.

   [6 marks]

   Total for question 4: 12 marks

End of questions