Equilibrium of a laminaEdexcel International A Level Further Maths: Subtopic test

1. 1
   A uniform rectangular lamina $ABCD$ has $AB=0.6$ m and $BC=0.4$ m, and mass $3$ kg. Take $g=9.8$ m s$^{-2}$. The lamina is freely suspended from a fixed point and hangs in equilibrium in a vertical plane.

   (a)

   The lamina is suspended from $A$. Find the angle that $AB$ makes with the vertical.

   [1 mark]

   • A$56.3^\circ$
   • B$26.6^\circ$
   • C$33.7^\circ$
   • D$36.9^\circ$

   (b)

   The lamina is now suspended from the midpoint $M$ of $AB$. What angle does $AB$ make with the vertical?

   [1 mark]

   • A$90^\circ$
   • B$0^\circ$
   • C$33.7^\circ$
   • D$45^\circ$

   (c)

   The lamina is suspended from $A$. Find the vertical distance of $G$ below $A$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform rectangular lamina $PQRS$ has $PQ=40$ cm, $QR=30$ cm and mass $2$ kg. It is free to rotate in a vertical plane about a smooth fixed horizontal axis through $P$, and hangs in equilibrium.

   (a)

   A particle of mass $1$ kg is attached to the lamina at $Q$. Find the distance of the centre of mass of the lamina and particle from $PS$.

   [1 mark]

   • A$20$ cm
   • B$30$ cm
   • C$40$ cm
   • D$26.7$ cm

   (b)

   With the particle of mass $1$ kg attached at $Q$, find the angle that $PQ$ makes with the vertical.

   [1 mark]

   • A$69.4^\circ$
   • B$20.6^\circ$
   • C$53.1^\circ$
   • D$14.0^\circ$

   (c)

   The particle is removed. Find the angle that $PQ$ makes with the vertical.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rectangular lamina $ABCD$ has $AB=30$ cm and $BC=50$ cm. It stands in a vertical plane with the edge $AB$ in contact with a rough plane inclined at an angle $\alpha$ to the horizontal. $AB$ lies along a line of greatest slope, with $A$ the lower end.

   (a)

   The lamina is on the point of toppling. Find the greatest value of $\alpha$.

   [3 marks]

   (b)

   The lamina is about to topple but is not sliding. Find the least possible coefficient of friction between the lamina and the plane.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform lamina is made from a square $ABCD$ of side $12$ cm and an isosceles triangle $CDE$ attached along $CD$, with $E$ outside the square at perpendicular distance $12$ cm from $CD$. The lamina hangs freely in equilibrium in a vertical plane, suspended from $A$.

   (a)

   Find the angle that $AB$ makes with the vertical.

   [6 marks]

   (b)

   The lamina has mass $9$ kg and a particle of mass $3$ kg is attached at $B$. Find the angle that $AB$ now makes with the vertical when the lamina hangs freely from $A$.

   [6 marks]

   Total for question 4: 12 marks

End of questions