Centre of mass of particles and composite bodiesAQA A-Level Further Maths: Subtopic test

1. 1
   Three particles of masses $2$ kg, $3$ kg and $5$ kg are at the points $(1,2)$, $(4,-1)$ and $(-2,3)$ respectively. All coordinates are in metres.

   (a)

   Find the $x$-coordinate of the centre of mass of the system.

   [1 mark]

   • A$0.4$ m
   • B$1.0$ m
   • C$4.0$ m
   • D$2.4$ m

   (b)

   Find the $y$-coordinate of the centre of mass of the system.

   [1 mark]

   • A$16$ m
   • B$1.3$ m
   • C$1.6$ m
   • D$2.2$ m

   (c)

   A fourth particle of mass $10$ kg is added so that the centre of mass of the four particles is at the origin. Find the coordinates of the fourth particle.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform lamina is L-shaped. It is made from a rectangle $8$ cm by $2$ cm and a rectangle $2$ cm by $4$ cm joined along an edge of length $2$ cm. With axes along the two outer edges meeting at the corner $O$, the first rectangle occupies $0\le x\le8$, $0\le y\le2$ and the second occupies $0\le x\le2$, $2\le y\le6$ (lengths in cm).

   (a)

   Find the $x$-coordinate of the centre of mass of the lamina.

   [1 mark]

   • A$2.5$ cm
   • B$3.0$ cm
   • C$2.0$ cm
   • D$4.0$ cm

   (b)

   Find the $y$-coordinate of the centre of mass of the lamina.

   [1 mark]

   • A$2.5$ cm
   • B$3.0$ cm
   • C$1.0$ cm
   • D$2.0$ cm

   (c)

   The lamina has mass $0.48$ kg. A particle of mass $0.12$ kg is attached at the point $(8,2)$. Find the $x$-coordinate of the centre of mass of the combined body.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform square lamina $ABCD$ has side $20$ cm. A circular hole of radius $4$ cm is cut from it. Take $A$ as the origin, with $AB$ along the $x$-axis and $AD$ along the $y$-axis, so that the centre of the hole is at $(14,10)$ and $C$ is at $(20,20)$ (lengths in cm).

   (a)

   Find the distance of the centre of mass of the lamina from $AD$.

   [3 marks]

   (b)

   The lamina has mass $0.5$ kg. A particle of mass $0.1$ kg is attached at $C$. Find the coordinates of the centre of mass of the combined body.
   Use your answer to (a).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform lamina is made by joining a rectangle $ABCD$ to an isosceles triangle $CDE$ along $CD$, with $E$ on the opposite side of $CD$ from the rectangle. $AB=12$ cm, $BC=8$ cm and the height of the triangle from $E$ to $CD$ is $6$ cm. Take $A$ as the origin, with $AB$ along the $x$-axis and $AD$ along the $y$-axis, so that $A(0,0)$, $B(12,0)$, $C(12,8)$, $D(0,8)$ and $E(6,14)$.

   (a)

   Find the distances of the centre of mass of the lamina from $AD$ and from $AB$.

   [6 marks]

   (b)

   A square of side $4$ cm is cut from the corner $A$, with sides along $AB$ and $AD$. Find the coordinates of the centre of mass of the remaining lamina.

   [6 marks]

   Total for question 4: 12 marks

End of questions