Mechanics 2 (M2): Centres of massEdexcel International A Level Further Maths: Topic test

1. 1
   Three particles of masses $2$ kg, $5$ kg and $3$ kg lie on the $x$-axis at the points with $x$-coordinates $-3$, $0$ and $4$ respectively, where the unit of length is the metre.

   (a)

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

   [1 mark]

   • A$6$ m
   • B$0.6$ m
   • C$\frac13$ m
   • D$-0.6$ m

   (b)

   A fourth particle is placed at $x=-6$ so that the centre of mass of the four particles is at the origin. Find the mass of the fourth particle.

   [1 mark]

   • A$0.6$ kg
   • B$1.2$ kg
   • C$6$ kg
   • D$1$ kg

   (c)

   The $5$ kg particle is removed. Find the $x$-coordinate of the centre of mass of the remaining two particles.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Particles of masses $3$ kg, $2$ kg and $5$ kg are placed at the points $A(2,5)$, $B(-4,1)$ and $C(6,-3)$ respectively, where the unit of length is the metre.

   (a)

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

   [1 mark]

   • A$2.8$ m
   • B$\frac43$ m
   • C$28$ m
   • D$0.2$ m

   (b)

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

   [1 mark]

   • A$2.8$ m
   • B$1$ m
   • C$0.2$ m
   • D$2$ m

   (c)

   A further particle of mass $10$ kg is placed at the origin. Find the coordinates of the centre of mass of the four particles.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform square lamina $ABCD$ has sides of length $24$ cm. A circular hole of radius $6$ cm is cut from the lamina so that the hole touches the sides $AB$ and $AD$.

   (a)

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

   [3 marks]

   (b)

   The lamina is freely suspended from the corner $B$ and hangs in equilibrium. Find the angle between $BA$ and the vertical.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform lamina of mass $2.4$ kg consists of a rectangle $ABCD$, with $AB=30$ cm and $AD=20$ cm, and a semicircle of diameter $AD$ attached outside the rectangle along $AD$. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the distance of the centre of mass of the lamina from $AD$, and state its distance from $AB$.

   [6 marks]

   (b)

   A particle of mass $m$ kg is attached to the lamina at the point $P$ on the semicircular edge that is furthest from $AD$. The lamina and particle are freely suspended from $D$ and hang in equilibrium with $AD$ vertical. Find $m$ and the magnitude of the force exerted on the lamina at $D$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A uniform triangular lamina has vertices at $(0,0)$, $(9,0)$ and $(3,6)$, where the unit of length is the centimetre.

   (a)

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

   [1 mark]

   • A$3$ cm
   • B$4.5$ cm
   • C$4$ cm
   • D$12$ cm

   (b)

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

   [1 mark]

   • A$2$ cm
   • B$3$ cm
   • C$6$ cm
   • D$4$ cm

   (c)

   A particle has the same mass as the lamina and is fixed at the point $(10,8)$. Find the coordinates of the centre of mass of the lamina and particle together.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A uniform triangular lamina $ABC$ has a right angle at $B$, with $AB=12$ cm and $BC=18$ cm. The lamina is freely suspended from a point and hangs in equilibrium in a vertical plane.

   (a)

   The lamina hangs freely from $A$. Find the angle between $AB$ and the vertical.

   [1 mark]

   • A$53.1^\circ$
   • B$56.3^\circ$
   • C$33.7^\circ$
   • D$36.9^\circ$

   (b)

   The lamina is now freely suspended from $C$. Find the angle between $CB$ and the vertical.

   [1 mark]

   • A$71.6^\circ$
   • B$18.4^\circ$
   • C$26.6^\circ$
   • D$33.7^\circ$

   (c)

   Find the distance $AG$, where $G$ is the centre of mass of the lamina.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A uniform rectangular lamina $ABCD$ has $AB=24$ cm, $BC=10$ cm and mass $5$ kg. A particle of mass $3$ kg is attached to the lamina at $C$.

   (a)

   Find the distance of the centre of mass of the combined body from $AD$ and from $AB$.

   [3 marks]

   (b)

   The combined body is freely suspended from $A$ and hangs in equilibrium. Find the angle between $AB$ and the vertical.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A uniform circular disc has centre $O$ and radius $10$ cm. A circular hole of radius $5$ cm is cut from the disc, with its centre $H$ at the midpoint of the radius $OX$, so that the hole touches the rim at $X$. The remaining lamina has mass $0.75$ kg.

   (a)

   Find the distance of the centre of mass of the lamina from $O$, and state where it lies.

   [6 marks]

   (b)

   The lamina is freely suspended from the point $P$ on the rim, where $OP$ is perpendicular to $OX$. Find the angle between $OP$ and the vertical. A particle of mass $m$ kg is then attached at $X$, and the lamina now hangs with $OP$ vertical. Find $m$.

   [6 marks]

   Total for question 8: 12 marks

End of questions