Moments and equilibrium of rigid bodiesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A uniform horizontal beam $AB$ of length $6$ m and mass $15$ kg rests on two smooth supports at $C$ and $D$, where $AC=1$ m and $DB=2$ m. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the reaction of the support at $C$.

   [1 mark]

   • A$98$ N
   • B$73.5$ N
   • C$49$ N
   • D$147$ N

   (b)

   A particle of mass $M$ kg is placed at $B$ and the beam is on the point of tilting about $D$. Find $M$.

   [1 mark]

   • A$3.75$
   • B$7.5$
   • C$15$
   • D$22.5$

   (c)

   A particle of mass $3$ kg is now placed at $A$ (and no particle at $B$). Find the magnitude of the reaction at $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform ladder $AB$ of mass $20$ kg and length $5$ m rests with end $A$ on rough horizontal ground and end $B$ against a smooth vertical wall. The end $B$ is $4$ m above the ground. The ladder is in equilibrium in a vertical plane perpendicular to the wall. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the normal reaction of the ground on the ladder.

   [1 mark]

   • A$98$ N
   • B$20$ N
   • C$73.5$ N
   • D$196$ N

   (b)

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

   [1 mark]

   • A$\frac38$
   • B$\frac34$
   • C$\frac83$
   • D$\frac35$

   (c)

   Find the magnitude of the total force exerted by the ground on the ladder.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rod $AB$ of mass $9$ kg and length $2$ m is smoothly hinged to a vertical wall at $A$. The rod is held in equilibrium in a horizontal position by a light string joining $B$ to a point $C$ on the wall, where $C$ is $1.5$ m vertically above $A$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform ladder $AB$ of mass $20$ kg and length $5$ m rests with end $A$ on rough horizontal ground and end $B$ against a rough vertical wall. The coefficient of friction is $\frac12$ between the ladder and the ground and also $\frac12$ between the ladder and the wall. The ladder is in a vertical plane perpendicular to the wall and makes an angle $\theta$ with the horizontal. Take $g=9.8$ m s$^{-2}$.

   (a)

   Given that the ladder is in limiting equilibrium with no one on it, find the value of $\theta$.

   [6 marks]

   (b)

   In a second situation, $\tan\theta=\frac43$ and a man of mass $40$ kg stands on the ladder at the point $C$. The ladder is about to slip. Find $AC$.

   [6 marks]

   Total for question 4: 12 marks

End of questions