Further Mechanics 2: Further centres of mass and equilibriumEdexcel A-Level Further Maths: Topic test

1. 1
   A uniform rectangular lamina $OABC$ has $OA=8$ cm and $OC=6$ cm. Cartesian axes $Ox$ and $Oy$ lie along $OA$ and $OC$ respectively. A rectangle measuring 4 cm by 3 cm is cut from the corner $B$, with its sides parallel to the sides of $OABC$, leaving a lamina $L$.

   (a)

   What is the $y$-coordinate of the centre of mass of the rectangle that is removed?

   [1 mark]

   • A$3$
   • B$4.5$
   • C$6$
   • D$1.5$

   (b)

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

   [1 mark]

   • A$4$
   • B$4.4$
   • C$10$
   • D$\frac{10}{3}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform rod $AB$ of mass 8 kg and length 3 m is freely hinged at $A$ to a vertical wall. The rod is held horizontal by a light string joining $B$ to the point $C$ on the wall, where $C$ is 2 m vertically above $A$. Take $g=9.8$ m s⁻².

   (a)

   What is the size of angle $ABC$?

   [1 mark]

   • A$33.7^\circ$
   • B$56.3^\circ$
   • C$41.8^\circ$
   • D$48.2^\circ$

   (b)

   What is the tension in the string?

   [1 mark]

   • A$39.2$ N
   • B$141$ N
   • C$70.7$ N
   • D$47.1$ N

   (c)

   Find the horizontal component of the force exerted on the rod by the hinge.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The region $R$ is bounded by the curve $y=4-x^2$, the $x$-axis and the $y$-axis, where $x\ge0$. A uniform lamina is cut in the shape of $R$ and has its centre of mass at $(\bar x,\bar y)$.

   (a)

   Show that $\bar x=\frac34$.

   [3 marks]

   (b)

   Show that $\bar y=\frac85$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform solid $S$ is made by attaching a uniform solid cylinder of radius 3 cm and height 8 cm to a uniform solid cone of base radius 3 cm and height 12 cm, both made of the same material. The plane face of the cone is joined to one plane face of the cylinder, so $S$ has a circular base of radius 3 cm formed by the other plane face of the cylinder.

   (a)

   Show that the centre of mass of $S$ is $\frac{19}{3}$ cm from its circular base.

   [6 marks]

   (b)

   $S$ is placed with its circular base on a rough plane inclined at angle $\alpha$ to the horizontal. The coefficient of friction between $S$ and the plane is 0.6 and $\alpha$ is gradually increased from zero. Determine whether $S$ topples or slips first, giving the value of $\alpha$ at which this happens.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A uniform circular disc has centre $O$ and radius 10 cm. A circular hole of radius 5 cm is cut from the disc so that the centre of the hole is 5 cm from $O$. The remaining lamina $L$ has centre of mass $G$.

   (a)

   What fraction of the mass of the original disc is removed?

   [1 mark]

   • A$\frac12$
   • B$\frac13$
   • C$\frac14$
   • D$\frac{1}{16}$

   (b)

   Let $\bar x$ be the distance $OG$, measured from $O$ on the side away from the centre of the hole. Which equation gives $\bar x$?

   [1 mark]

   • A$\frac34\bar x=\frac14\times5$
   • B$\bar x=\frac14\times5$
   • C$\frac34\bar x=\frac14\times10$
   • D$\frac14\bar x=\frac34\times5$

   (c)

   $L$ is freely suspended from the point $P$ on its rim that is furthest from the centre of the hole. Find the distance $PG$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A uniform beam $AB$ of mass 40 kg and length 6 m rests horizontally on two smooth supports at $C$ and $D$, where $AC=1$ m and $DB=1$ m. A man of mass 90 kg stands on the beam. Take $g=9.8$ m s⁻².

   (a)

   The man stands at the midpoint of the beam. What is the magnitude of the reaction at $C$?

   [1 mark]

   • A$1274$ N
   • B$392$ N
   • C$882$ N
   • D$637$ N

   (b)

   The man now walks from the midpoint towards $B$, beyond $D$. How far is he from $D$ when the reaction at $C$ first becomes zero?

   [1 mark]

   • A$0.444$ m
   • B$0.889$ m
   • C$1.125$ m
   • D$2.22$ m

   (c)

   The man stands 0.5 m from $B$. Find the magnitude of the reaction at $C$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The region $R$ is bounded by the curve $y=x^2$, the $x$-axis and the line $x=2$. A uniform solid $S$ is formed by rotating $R$ through $2\pi$ radians about the $x$-axis, so $S$ has its vertex at $O$ and a plane circular face where $x=2$.

   (a)

   Show that the centre of mass of $S$ is $\frac53$ from $O$.

   [3 marks]

   (b)

   $S$ is freely suspended from a point $A$ on the circumference of its plane circular face. Find the angle between the plane face and the vertical.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A uniform L-shaped lamina of mass 4.8 kg has vertices at $(0,0)$, $(9,0)$, $(9,3)$, $(3,3)$, $(3,10)$ and $(0,10)$, with coordinates in centimetres relative to horizontal and vertical axes $Ox$ and $Oy$. The lamina lies in a vertical plane. Take $g=9.8$ m s⁻².

   (a)

   Find the coordinates of the centre of mass of the lamina.

   [6 marks]

   (b)

   The lamina is freely suspended from the vertex $A(9,0)$ and is held in equilibrium, with $Ox$ horizontal, by a horizontal force of magnitude $P$ newtons, in the positive $x$-direction, acting at the vertex $(0,10)$ in the plane of the lamina. Find the value of $P$ and the magnitude of the force acting on the lamina at $A$.

   [6 marks]

   Total for question 8: 12 marks

End of questions