Further Mechanics 1: Centres of mass and momentsAQA A-Level Further Maths: Topic test

1. 1
   A light rod $AB$ of length $1.2$ m carries three particles: one of mass $0.5$ kg at $A$, one of mass $0.3$ kg at the point $C$ on the rod where $AC=0.4$ m, and one of mass $0.2$ kg at $B$.

   (a)

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

   [1 mark]

   • A$0.36$ m
   • B$0.53$ m
   • C$0.60$ m
   • D$0.84$ m

   (b)

   A fourth particle of mass $0.5$ kg is now attached to the rod at $A$. How far from $A$ is the centre of mass of the four particles?

   [1 mark]

   • A$0.36$ m
   • B$0.24$ m
   • C$0.54$ m
   • D$0.18$ m

   (c)

   Instead of the fourth particle in part (b), a particle of mass $M$ kg is attached to the rod at $B$ (in addition to the particle already there). The centre of mass of the system is then at the midpoint of $AB$. Find $M$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A uniform beam $AB$ of length $4$ m and mass $24$ kg is held in equilibrium in a horizontal position. The end $A$ is smoothly hinged to a vertical wall. A light rope joins $B$ to a point $C$ on the wall, where $C$ is $3$ m vertically above $A$.

   (a)

   What is the tension in the rope?

   [1 mark]

   • A$117.6$ N
   • B$147$ N
   • C$196$ N
   • D$235.2$ N

   (b)

   What is the magnitude of the horizontal component of the force exerted by the hinge on the beam?

   [1 mark]

   • A$117.6$ N
   • B$196$ N
   • C$0$ N
   • D$156.8$ N

   (c)

   Find the vertical component of the force exerted by the hinge on the beam, and state its direction.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform lamina occupies the region $R$ bounded by the curve $y=x^{3}$, the $x$-axis and the line $x=2$. Lengths are measured in centimetres.

   (a)

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

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The region bounded by the curve $y=\sqrt{x}$, the $x$-axis and the line $x=9$ is rotated through $2\pi$ radians about the $x$-axis to form a uniform solid $S$ of mass $1.2$ kg. Lengths are measured in centimetres, and the vertex of $S$ is at the origin $O$.

   (a)

   Show that the centre of mass of $S$ lies on the axis of symmetry at a distance of $6$ cm from $O$.

   [6 marks]

   (b)

   The solid $S$ is placed with its circular face on a rough horizontal table, so that $O$ is $9$ cm above the table. A horizontal force $P$, perpendicular to the axis of $S$, is applied at $O$ and is slowly increased from zero until $S$ is about to topple. Find the value of $P$ at this instant and the least value of the coefficient of friction between $S$ and the table for $S$ to topple rather than slip.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A uniform T-shaped lamina is made from a horizontal bar, which is a rectangle $12$ cm by $3$ cm, and a vertical stem, which is a rectangle $3$ cm by $9$ cm. The top edge of the stem is joined to the bottom edge of the bar with the stem centred on the bar, so the lamina is symmetrical about a vertical line and is $12$ cm high. The lamina has mass $0.63$ kg.

   (a)

   How far below the top edge of the bar is the centre of mass of the lamina?

   [1 mark]

   • A$4.50$ cm
   • B$6.00$ cm
   • C$7.93$ cm
   • D$4.07$ cm

   (b)

   The lamina hangs freely in equilibrium from the left-hand end $A$ of the top edge of the bar. What angle does the top edge make with the horizontal?

   [1 mark]

   • A$34.2^{\circ}$
   • B$55.8^{\circ}$
   • C$36.9^{\circ}$
   • D$18.7^{\circ}$

   (c)

   A particle is attached to the midpoint of the bottom edge of the stem so that the centre of the whole body is $6$ cm below the top edge of the bar. Find the mass of the particle.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A uniform solid is formed by rotating about the $x$-axis the region bounded by the curve $y=x^{2}$, the $x$-axis and the line $x=3$. Lengths are measured in centimetres.

   (a)

   Which expression gives the volume of the solid?

   [1 mark]

   • A$\pi\displaystyle\int_0^3x^{2}\,dx$
   • B$\displaystyle\int_0^3x^{4}\,dx$
   • C$\pi\displaystyle\int_0^3x^{4}\,dx$
   • D$\pi\displaystyle\int_0^9x^{4}\,dx$

   (b)

   How far from the origin, along the $x$-axis, is the centre of mass of the solid?

   [1 mark]

   • A$2.25$ cm
   • B$2.5$ cm
   • C$1.5$ cm
   • D$2.0$ cm

   (c)

   Find the exact volume of the solid.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A uniform triangular lamina $ABC$ has a right angle at $A$, with $AB=0.6$ m and $AC=0.8$ m. Take $A$ as the origin, with $AB$ along the $x$-axis and $AC$ along the $y$-axis.

   (a)

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

   [3 marks]

   (b)

   The lamina hangs freely in equilibrium from the vertex $B$. Find the angle between $AB$ and the vertical.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A toy is made by joining a uniform solid hemisphere of radius $4$ cm to one end of a uniform solid cylinder of radius $4$ cm and height $10$ cm. Both parts are made of the same material. The plane face of the hemisphere coincides with a circular end of the cylinder, so the toy has an axis of symmetry.

   (a)

   The centre of mass of a uniform solid hemisphere of radius $r$ is $\tfrac38r$ from the centre of its plane face, and its volume is $\tfrac23\pi r^{3}$. Show that the centre of mass of the toy is $3.63$ cm from the plane face where the two parts are joined.

   [6 marks]

   (b)

   The toy is placed with the flat circular end of the cylinder on a rough plane inclined at an angle $\alpha$ to the horizontal, and it does not slip. Find the greatest value of $\alpha$ for which the toy does not topple, and the least coefficient of friction needed to prevent slipping when $\alpha$ has this value.

   [6 marks]

   Total for question 8: 12 marks

End of questions