Mechanics 2 (M2): Statics of rigid bodiesEdexcel International A Level Further Maths: Topic test

1. 1
   A light rod $AB$ of length $3$ m carries a particle of mass $6$ kg at $A$ and a particle of mass $M$ kg at $B$. The rod rests horizontally in equilibrium on a smooth pivot at the point $C$, where $AC=1$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the value of $M$.

   [1 mark]

   • A$3$
   • B$12$
   • C$6$
   • D$4$

   (b)

   Find the magnitude of the force exerted on the rod by the pivot, in N.

   [1 mark]

   • A$9$
   • B$58.8$
   • C$29.4$
   • D$88.2$

   (c)

   The mass at $B$ is $3$ kg. A further particle of mass $3$ kg is attached to the rod at its midpoint. The pivot is moved to a point $D$ so that the rod rests horizontally in equilibrium. Find $AD$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform rod $PQ$ of mass $12$ kg and length $3$ m hangs horizontally in equilibrium, supported by two vertical light ropes. One rope is attached at $P$ and the other at the point $R$ of the rod, where $PR=2$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the rope attached at $R$, in N.

   [1 mark]

   • A$29.4$
   • B$58.8$
   • C$117.6$
   • D$88.2$

   (b)

   Find the tension in the rope attached at $P$, in N.

   [1 mark]

   • A$88.2$
   • B$58.8$
   • C$29.4$
   • D$117.6$

   (c)

   A block of mass $3$ kg is placed on the rod at $Q$. Find the tension in the rope attached at $P$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform ladder $AB$ of mass $20$ kg and length $6$ m rests in equilibrium in a vertical plane perpendicular to a smooth vertical wall. End $A$ is on rough horizontal ground and end $B$ is against the wall. The ladder makes an angle $\theta$ with the horizontal, where $\tan\theta=2$. A man of mass $45$ kg stands on the ladder at the point $C$, where $AC=4$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the normal reaction between the ladder and the wall.

   [3 marks]

   (b)

   The ladder is on the point of slipping. Find the coefficient of friction between the ladder and the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform rod $AB$ of mass $10$ kg and length $6$ m rests in equilibrium in a vertical plane, with its end $A$ on rough horizontal ground. The rod rests on a smooth horizontal peg $C$ and makes an angle $\alpha$ with the horizontal, where $\tan\alpha=\frac34$. The distance $AC$ is $4$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   (i) Find the magnitude of the reaction of the peg on the rod.
   (ii) Find the magnitude of the frictional force acting on the rod at $A$.

   [6 marks]

   (b)

   (i) Find the magnitude of the normal reaction of the ground on the rod at $A$.
   (ii) The coefficient of friction between the rod and the ground is $0.6$. Determine, with justification, whether the rod can remain in equilibrium.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A uniform beam $AB$ of mass $40$ kg and length $5$ m rests horizontally on two supports, one at $A$ and one at the point $C$ of the beam, where $AC=4$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the reaction at $A$, in N.

   [1 mark]

   • A$245$
   • B$147$
   • C$196$
   • D$392$

   (b)

   A particle of mass $m$ kg is placed at $B$. Find the greatest value of $m$ for which the beam stays in equilibrium resting on both supports, in kg.

   [1 mark]

   • A$100$
   • B$40$
   • C$60$
   • D$24$

   (c)

   A particle of mass $15$ kg is placed at $B$. Find the magnitude of the reaction at $A$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A uniform ladder $AB$ of mass $10$ kg and length $5$ m rests in a vertical plane with its end $A$ on smooth horizontal ground and its end $B$ against a smooth vertical wall. The ladder is held in equilibrium by a horizontal rope joining $A$ to the base of the wall, and makes an angle $\theta$ with the ground, where $\tan\theta=\frac43$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the rope, in N.

   [1 mark]

   • A$65.3$
   • B$98$
   • C$61.25$
   • D$36.75$

   (b)

   Find the magnitude of the normal reaction of the ground on the ladder, in N.

   [1 mark]

   • A$98$
   • B$134.75$
   • C$78.4$
   • D$58.8$

   (c)

   The rope will break if its tension exceeds $45$ N. Find the smallest angle that the ladder can make with the ground, in degrees to 1 decimal place, without the rope breaking.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A uniform rod $AB$ of mass $6$ kg and length $2$ m is smoothly hinged to a fixed point at $A$. The rod is held in equilibrium at an angle $\theta$ above the horizontal, where $\tan\theta=\frac34$, by a light string attached to $B$. The string is perpendicular to the rod and lies in the same vertical plane as the rod, with its other end fixed at a point above the rod. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A uniform ladder $AB$ of mass $14$ kg and length $6$ m rests in a vertical plane perpendicular to a rough vertical wall, with its end $A$ on rough horizontal ground and its end $B$ against the wall. The coefficient of friction between the ladder and the ground is $\frac12$ and between the ladder and the wall is $\frac13$. The ladder makes an angle $\theta$ with the horizontal and is in limiting equilibrium, with friction limiting at both $A$ and $B$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the normal reaction at $A$ and the magnitude of the normal reaction at $B$.

   [6 marks]

   (b)

   Find the value of $\theta$, in degrees to 1 decimal place.

   [6 marks]

   Total for question 8: 12 marks

End of questions