M1: MomentsEdexcel International A Level Maths: Topic test

1. 1
   A uniform seesaw plank $AB$ of length $5$ m and mass $40$ kg is pivoted at its midpoint $C$ and is horizontal. A child of mass $30$ kg sits at $A$. Take $g=9.8$ m s⁻².

   (a)

   Find the moment of the weight of the child about $C$.

   [1 mark]

   • A$75$ N m
   • B$1470$ N m
   • C$367.5$ N m
   • D$735$ N m

   (b)

   A second child of mass $50$ kg sits on $CB$ so that the plank remains in equilibrium, horizontal. Find the distance of the second child from $C$.

   [1 mark]

   • A$2.5$ m
   • B$1.5$ m
   • C$4.17$ m
   • D$1.67$ m

   (c)

   With both children on the plank, find the magnitude of the force exerted by the pivot on the plank.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform metre rule of mass $0.12$ kg is balanced horizontally on a pivot at the $30$ cm mark by a mass $M$ kg hung from the $0$ cm end. Take $g=9.8$ m s⁻².

   (a)

   Find the value of $M$.

   [1 mark]

   • A$0.08$
   • B$0.18$
   • C$0.20$
   • D$0.12$

   (b)

   Find the magnitude of the force exerted by the pivot on the rule.

   [1 mark]

   • A$1.18$ N
   • B$0.78$ N
   • C$1.96$ N
   • D$3.92$ N

   (c)

   The pivot is moved to the $40$ cm mark. The mass $M$ stays at the $0$ cm mark and a mass $m$ kg is hung from the $90$ cm mark so that the rule balances horizontally. Find $m$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rigid lever $AB$ of length $1.5$ m is pivoted at a point $C$, where $AC=0.3$ m. A boulder rests on the end $A$ and exerts a downward force of $2400$ N on the lever. A worker pushes vertically downwards on the end $B$ with a force of magnitude $F$ N so that the lever is in equilibrium in a horizontal position. Take $g=9.8$ m s⁻².

   (a)

   Modelling the lever as light, find the value of $F$.

   [3 marks]

   (b)

   In fact the lever is uniform and has mass $20$ kg. Find the new value of $F$ and the magnitude of the force exerted by the pivot on the lever.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform girder $AB$ of length $12$ m and mass $150$ kg rests horizontally on two supports at $P$ and $Q$, where $AP=2$ m and $QB=4$ m. A crate of mass $30$ kg is placed on the girder at $B$. Take $g=9.8$ m s⁻².

   (a)

   Find the magnitude of the reaction at each support.

   [6 marks]

   (b)

   The crate at $B$ is replaced by a crate of mass $M$ kg. Find the greatest value of $M$ for which the girder remains in equilibrium in a horizontal position, and the magnitude of the reaction at $Q$ in this case.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A uniform horizontal beam $AB$ of length $2.4$ m and mass $9$ kg is smoothly hinged to a wall at $A$ and supported by a vertical cable attached at $B$, so that the force at the hinge acts vertically. A block of mass $6$ kg is attached to the beam at $B$. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in the cable.

   [1 mark]

   • A$44.1$ N
   • B$102.9$ N
   • C$58.8$ N
   • D$147$ N

   (b)

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

   [1 mark]

   • A$102.9$ N
   • B$147$ N
   • C$249.9$ N
   • D$44.1$ N

   (c)

   The cable breaks if its tension exceeds $150$ N. The block at $B$ is replaced by a block of mass $m$ kg. Find the greatest value of $m$ for which the cable does not break.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A non-uniform rod $AB$ of length $2$ m and mass $6$ kg balances horizontally on a pivot at a point $0.8$ m from $A$ when a particle of mass $4$ kg is attached to the rod at $A$. Take $g=9.8$ m s⁻².

   (a)

   Find the distance of the centre of mass of the rod from $A$.

   [1 mark]

   • A$0.53$ m
   • B$1.00$ m
   • C$1.33$ m
   • D$0.80$ m

   (b)

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

   [1 mark]

   • A$98$ N
   • B$58.8$ N
   • C$39.2$ N
   • D$196$ N

   (c)

   The particle at $A$ is replaced by a particle of mass $2$ kg, also at $A$. Find the distance from $A$ of the point at which the rod must be pivoted to balance horizontally.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A light plank $AB$ of length $3$ m rests horizontally on the two banks of a stream, at $A$ and at $B$. A girl of mass $40$ kg stands on the plank at a point $1$ m from $A$. Take $g=9.8$ m s⁻².

   (a)

   Find the magnitude of the reaction of the bank on the plank at each end.

   [3 marks]

   (b)

   A boy of mass $25$ kg then stands on the plank, $x$ m from $A$, so that the reactions at $A$ and $B$ are equal in magnitude. Find the value of $x$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A uniform diving board $AB$ of length $3$ m and mass $36$ kg is horizontal. It rests on a support at $C$, where $AC=1$ m, and is held down by a bolt at $A$ that exerts a vertical downward force on the board. A diver of mass $45$ kg stands on the board at $B$. Take $g=9.8$ m s⁻².

   (a)

   Find the magnitude of the force exerted by the bolt on the board and the magnitude of the reaction at $C$.

   [6 marks]

   (b)

   The bolt will fail if the force it exerts exceeds $800$ N. The diver instead stands at a point on $CB$ at a distance $d$ m from $C$. Find the greatest value of $d$ for which the bolt does not fail, and the reaction at $C$ in this case.

   [6 marks]

   Total for question 8: 12 marks

End of questions