Mechanics 3 (M3): Statics of rigid bodiesEdexcel International A Level Further Maths: Topic test

1. 1
   A uniform lamina occupies the region $R$ bounded by the curve $y=4-x^2$ and the $x$-axis.

   (a)

   Find the area of $R$.

   [1 mark]

   • A$\frac{16}{3}$
   • B$8$
   • C$\frac{32}{3}$
   • D$\frac{64}{3}$

   (b)

   Which expression gives the $y$-coordinate $\bar y$ of the centre of mass of the lamina?

   [1 mark]

   • A$\dfrac{\frac12\int_{-2}^{2}(4-x^2)^2\,\mathrm{d}x}{\int_{-2}^{2}(4-x^2)\,\mathrm{d}x}$
   • B$\dfrac{\int_{-2}^{2}(4-x^2)^2\,\mathrm{d}x}{\int_{-2}^{2}(4-x^2)\,\mathrm{d}x}$
   • C$\dfrac{\int_{-2}^{2}x(4-x^2)\,\mathrm{d}x}{\int_{-2}^{2}(4-x^2)\,\mathrm{d}x}$
   • D$\dfrac{\frac12\int_{-2}^{2}(4-x^2)\,\mathrm{d}x}{\int_{-2}^{2}(4-x^2)^2\,\mathrm{d}x}$

   (c)

   Show that $\bar y=\frac85$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform rectangular lamina $ABCD$ has $AB=8$ cm, $BC=6$ cm and mass $2.4$ kg. It hangs in equilibrium, freely suspended from the vertex $A$.

   (a)

   Find the angle that $AB$ makes with the vertical.

   [1 mark]

   • A$53.1^\circ$
   • B$36.9^\circ$
   • C$45^\circ$
   • D$73.7^\circ$

   (b)

   Find the distance $AG$ from the vertex $A$ to the centre of mass $G$, in cm.

   [1 mark]

   • A$7$
   • B$10$
   • C$3.5$
   • D$5$

   (c)

   Find the magnitude of the force exerted on the lamina at $A$. Take $g=9.8$ m s$^{-2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform rod $AB$ of mass $5$ kg and length $1.6$ m has its end $A$ smoothly hinged to a vertical wall. The rod is held in equilibrium in a horizontal position by a light inextensible string. One end of the string is attached to $B$ and the other end is attached to a point $C$ on the wall, $1.2$ m vertically above $A$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string.

   [3 marks]

   (b)

   Find the magnitude and direction of the force exerted on the rod by the hinge.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform lamina $L$ is bounded by the line $y=x$ and the curve $y=\frac{x^2}{20}$, for $0\le x\le20$, where $x$ and $y$ are measured in centimetres. The origin $O$ and the point $A(20,20)$ are vertices of $L$.

   (a)

   Show that the centre of mass of $L$ is at the point $(10,8)$.

   [6 marks]

   (b)

   The lamina has mass $0.5$ kg. A particle of mass $0.3$ kg is attached to the lamina at $A$, and the lamina is freely suspended from $O$ and hangs in equilibrium. Find the angle that $OA$ makes with the downward vertical.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A uniform solid cone is formed by rotating through $360^\circ$ about the $x$-axis the region bounded by the line $y=\frac x2$, the $x$-axis and the line $x=12$, where lengths are in centimetres. The vertex of the cone is at the origin $O$.

   (a)

   Find the volume of the cone, in cm$^3$.

   [1 mark]

   • A$144\pi$
   • B$432\pi$
   • C$12\pi$
   • D$576\pi$

   (b)

   A uniform solid cylinder of the same material, of radius $6$ cm and height $2$ cm, is attached to the circular base of the cone, with the same axis. Find the distance of the centre of mass of the combined solid from $O$, in cm.

   [1 mark]

   • A$11$
   • B$10.3$
   • C$13$
   • D$9$

   (c)

   Show that the centre of mass of the cone alone is $9$ cm from $O$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A uniform rectangular block of mass $5$ kg has a square base of side $0.3$ m and a height of $0.8$ m. The block rests with one of its rectangular faces on a rough plane inclined at an angle $\alpha$ to the horizontal. The longer edges of the block are perpendicular to the plane. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the least value of $\alpha$ at which the block would topple, assuming that it does not slide.

   [1 mark]

   • A$69.4^\circ$
   • B$36.9^\circ$
   • C$10.6^\circ$
   • D$20.6^\circ$

   (b)

   Given that $\alpha=20^\circ$ and the block is in equilibrium, find the least possible value of the coefficient of friction between the block and the plane.

   [1 mark]

   • A$0.342$
   • B$0.940$
   • C$0.364$
   • D$2.75$

   (c)

   Given that $\alpha=20^\circ$ and the block is in equilibrium, find the magnitude of the frictional force on the block.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A uniform rod $AB$ of mass $10$ kg and length $4$ m rests in equilibrium with its end $A$ on rough horizontal ground. The rod rests against a smooth fixed peg $C$, where $AC=2.5$ m, and is inclined at $60^\circ$ to the horizontal. The rod lies in a vertical plane perpendicular to the peg. Take $g=9.8$ m s$^{-2}$.

   (a)

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

   [3 marks]

   (b)

   Find the least possible value of the coefficient of friction between the rod and the ground.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A uniform lamina is formed by joining a rectangle $ABCD$, with $AB=12$ cm and $BC=8$ cm, to a semicircle with diameter $BC$ and radius $4$ cm, lying outside the rectangle. The lamina has mass $1.2$ kg.

   (a)

   Show that the distance of the centre of mass of the lamina from $AD$ is $7.60$ cm, to $3$ significant figures.

   [6 marks]

   (b)

   The lamina is freely suspended from $C$. A particle of mass $m$ kg is attached to the lamina at the point $E$ of the semicircle that is furthest from $BC$, so that $CD$ hangs horizontally in equilibrium. Find $m$, and find the magnitude of the force exerted by the suspension on the lamina. Take $g=9.8$ m s$^{-2}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions