Centre of mass of discrete massesEdexcel International A Level Further Maths: Subtopic test

1. 1
   Particles of mass $2$ kg, $3$ kg and $5$ kg are placed on a straight horizontal line at distances $1$ m, $4$ m and $7$ m respectively from a fixed point $O$, all on the same side of $O$.

   (a)

   How far is the centre of mass of the three particles from $O$?

   [1 mark]

   • A$4.9$ m
   • B$4.0$ m
   • C$49$ m
   • D$3.1$ m

   (b)

   A fourth particle of mass $10$ kg is now placed $9$ m from $O$, on the same line and side. How far from $O$ is the new centre of mass of the four particles?

   [1 mark]

   • A$4.9$ m
   • B$9$ m
   • C$13.9$ m
   • D$6.95$ m

   (c)

   Instead, a fourth particle of mass $10$ kg is placed at a distance $d$ m from $O$ so that the centre of mass of the four particles is $6$ m from $O$. Find $d$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Particles of mass $2$ kg, $3$ kg and $5$ kg are placed at the points $(1,4)$, $(4,0)$ and $(3,2)$ respectively, where the coordinates are in metres relative to an origin $O$.

   (a)

   What is the $x$-coordinate of the centre of mass?

   [1 mark]

   • A$2.67$
   • B$29$
   • C$2.9$
   • D$2.3$

   (b)

   What is the $y$-coordinate of the centre of mass?

   [1 mark]

   • A$2.0$
   • B$1.8$
   • C$18$
   • D$2.4$

   (c)

   Find the distance of the centre of mass from the origin $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A light rod $AB$ has length $4$ m. Particles of mass $3$ kg, $m$ kg and $5$ kg are attached to the rod at $A$, at the midpoint of $AB$ and at $B$ respectively. The centre of mass of these three particles is $2.4$ m from $A$.

   (a)

   Find the value of $m$.

   [3 marks]

   (b)

   Given that $m=2$, a further particle of mass $2$ kg is attached to the rod so that the centre of mass of the four particles is $2.5$ m from $A$. Find the distance from $A$ at which it is attached.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Particles of mass $2$ kg, $4$ kg, $m$ kg and $3$ kg are placed at the points $A(0,0)$, $B(6,0)$, $C(6,4)$ and $D(0,4)$ respectively, with coordinates in metres. The $x$-coordinate of the centre of mass of the four particles is $3.5$.

   (a)

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

   [6 marks]

   (b)

   Given that $m=3$, the particle at $D$ is removed. Find the coordinates of the centre of mass of the remaining three particles, and the distance of this centre of mass from $B$.

   [6 marks]

   Total for question 4: 12 marks

End of questions