Moments and centres of mass of discrete massesEdexcel A-Level Further Maths: Subtopic test

1. 1
   Three particles of masses 2 kg, 3 kg and 5 kg are placed on a straight line at the points $x=1$ m, $x=4$ m and $x=9$ m respectively, where $x$ is the distance from a fixed origin $O$ on the line. Take $g=9.8$ m s⁻².

   (a)

   Find the distance of the centre of mass of the three particles from $O$.

   [1 mark]

   • A$4.67$ m
   • B$5.0$ m
   • C$59$ m
   • D$5.9$ m

   (b)

   A fourth particle of mass 10 kg is placed on the line at $x=8$ m. Find the distance of the new centre of mass from $O$.

   [1 mark]

   • A$13.9$ m
   • B$6.95$ m
   • C$5.5$ m
   • D$5.9$ m

   (c)

   Find the total moment of the weights of the three particles about $O$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Four particles of masses 1 kg, 2 kg, 3 kg and 4 kg are placed at the points with coordinates $(0,0)$, $(5,0)$, $(4,3)$ and $(1,3)$ respectively, where distances are in metres.

   (a)

   Find the $x$-coordinate of the centre of mass of the four particles.

   [1 mark]

   • A$2.5$ m
   • B$26$ m
   • C$2.6$ m
   • D$6.5$ m

   (b)

   Find the $y$-coordinate of the centre of mass of the four particles.

   [1 mark]

   • A$2.1$ m
   • B$1.5$ m
   • C$21$ m
   • D$5.25$ m

   (c)

   Find the distance between the centre of mass and the 2 kg particle.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rod $AB$ has length 6 m and mass 12 kg. It rests horizontally 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 magnitudes of the reactions at $C$ and $D$.

   [3 marks]

   (b)

   A particle of mass $m$ kg is placed on the rod at $B$ and the rod is on the point of tilting about $D$. Find $m$ and the reaction at $D$ in this case.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Four particles of masses 2 kg, 3 kg, $m$ kg and 5 kg are fixed to a light rigid plate at the points $A(-2,1)$, $B(3,4)$, $C(5,-2)$ and $D(-1,-3)$ respectively, where distances are in metres. The $x$-coordinate of the centre of mass of the four particles is 2.5.

   (a)

   Find the value of $m$ and the $y$-coordinate of the centre of mass.

   [6 marks]

   (b)

   The particle at $C$ is removed and a particle of mass $M$ kg is fixed to the plate at $E(4,6)$. The centre of mass of the new system of four particles lies on the line $y=x$. Find $M$.

   [6 marks]

   Total for question 4: 12 marks

End of questions