Equilibrium of rigid bodiesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A uniform rod $AB$ of mass $6$ kg and length $2$ m is freely hinged to a fixed point at $A$. A horizontal force of magnitude $P$ newtons is applied at $B$, and the rod rests in equilibrium inclined at an angle $\theta$ to the downward vertical, where $\tan\theta=\frac12$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the value of $P$.

   [1 mark]

   • A$29.4$
   • B$7.35$
   • C$117.6$
   • D$14.7$

   (b)

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

   [1 mark]

   • A$58.8$ N
   • B$73.5$ N
   • C$60.6$ N
   • D$14.7$ N

   (c)

   Find the angle that the force exerted by the hinge makes with the vertical.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform solid cylinder of radius $0.2$ m and height $0.6$ m is placed with its circular base on a plane inclined at an angle $\theta$ to the horizontal. The plane is rough. The angle $\theta$ is increased slowly from zero.

   (a)

   Find the value of $\tan\theta$ at which the cylinder is about to topple.

   [1 mark]

   • A$\frac13$
   • B$\frac23$
   • C$\frac32$
   • D$\frac43$

   (b)

   The coefficient of friction between the cylinder and the plane is $\mu$. Which condition on $\mu$ ensures that the cylinder topples without slipping?

   [1 mark]

   • A$\mu>\frac23$
   • B$\mu<\frac23$
   • C$\mu>\frac32$
   • D$\mu<\frac32$

   (c)

   The coefficient of friction between the cylinder and the plane is $0.5$. Find the angle at which the cylinder first begins to move, and state how it moves.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform ladder $AB$ of mass $20$ kg and length $5$ m rests in equilibrium with its end $A$ on rough horizontal ground and its end $B$ against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall, and makes an angle of $60^\circ$ with the horizontal. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the frictional force at $A$.

   [3 marks]

   (b)

   The coefficient of friction between the ladder and the ground is $0.4$. A man of mass $80$ kg climbs the ladder. Find the greatest distance along the ladder, from $A$, to which he can climb before the ladder slips.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform triangular lamina $ABC$ is right-angled at $B$, with $AB=12$ cm and $BC=9$ cm. The lamina hangs freely in equilibrium in a vertical plane, suspended from a point of the lamina.

   (a)

   Find the angle that $AB$ makes with the vertical when the lamina is suspended from $A$, and the angle that $BC$ makes with the vertical when it is suspended from $C$.

   [6 marks]

   (b)

   The lamina has mass $0.6$ kg. A particle of mass $0.3$ kg is attached to the lamina at $C$ and the lamina is freely suspended from $A$. Find the angle that $AB$ makes with the vertical.

   [6 marks]

   Total for question 4: 12 marks

End of questions