Further Mechanics 2: Centres of mass of plane figuresEdexcel A-Level Further Maths: Topic test

20 questions, 54 marks

Edexcel A-Level Further Maths

Further Mechanics 2: Centres of mass of plane figures topic test

Total 54 marks

Name

Class

Date

1
A light rigid rod $PQ$ has length $2.5$ m. Particles of mass $4$ kg and $6$ kg are attached to the rod at $P$ and $Q$ respectively.
(a)
How far from $P$ is the centre of mass of the two particles?
[1 mark]
A $1.5$ m
B $1.0$ m
C $1.25$ m
D $0.6$ m
(b)
A third particle of mass $10$ kg is attached to the rod at its midpoint. How far from $P$ is the centre of mass of the three particles?
[1 mark]
A $1.5$ m
B $1.25$ m
C $1.375$ m
D $1.125$ m
(c)
The rod, carrying all three particles, is smoothly pivoted at $P$ and held horizontal by a light vertical string attached at $Q$ . Find the tension in the string.
[2 marks]
Total for question 1: 4 marks
2
Three particles of masses $3$ kg, $2$ kg and $5$ kg are attached to a light rigid lamina at the points $P(2,1)$ , $Q(6,4)$ and $R(4,5)$ respectively, where the coordinates are in centimetres.
(a)
What is the $x$ -coordinate of the centre of mass of the three particles?
[1 mark]
A $4.0$ cm
B $3.8$ cm
C $3.4$ cm
D $3.2$ cm
(b)
What is the $y$ -coordinate of the centre of mass of the three particles?
[1 mark]
A $3.8$ cm
B $3.33$ cm
C $2.8$ cm
D $3.6$ cm
(c)
A fourth particle of mass $2$ kg is attached to the lamina so that the centre of mass of the four particles is at $(4,4)$ . Find the coordinates of the fourth particle.
[2 marks]
Total for question 2: 4 marks
3
A uniform circular lamina has centre $O$ and radius $8$ cm. A circular hole of radius $3$ cm is cut from the lamina, with centre $H$ , where $OH=4$ cm. The remaining shape is a lamina $L$ .
(a)
Find the distance of the centre of mass of $L$ from $O$ .
[3 marks]
(b)
The lamina $L$ is freely suspended from a point $P$ on its rim, where $OP$ is perpendicular to $OH$ . Find the angle between $OP$ and the vertical.
[4 marks]
Total for question 3: 7 marks
4
A uniform lamina is formed by joining a rectangle $ABCD$ , with $AB=12$ cm and $BC=8$ cm, to an isosceles triangle $DCE$ along the side $DC$ . The perpendicular distance from $E$ to $DC$ is $9$ cm, so that the lamina is a pentagon $ABCED$ .
(a)
Find the distance of the centre of mass of the lamina from $AB$ . The lamina is then freely suspended from $A$ . Find the angle between $AB$ and the vertical.
[6 marks]
(b)
The lamina has mass $3$ kg. A particle of mass $1$ kg is attached to the lamina at $A$ . The combined body is freely suspended from $B$ . Find the angle between $BA$ and the vertical.
[6 marks]
Total for question 4: 12 marks
5
A uniform beam $AB$ of mass $40$ kg and length $4$ m rests horizontally. It is smoothly hinged to a wall at $A$ and is held in equilibrium by a light vertical rope, attached to the beam at the point $C$ where $AC=3$ m and to a fixed point vertically above $C$ . Take $g=9.8$ m s $^{-2}$ .
(a)
What is the tension in the rope?
[1 mark]
A $196$ N
B $392$ N
C $523$ N
D $261$ N
(b)
What is the magnitude of the vertical force exerted on the beam by the hinge?
[1 mark]
A $131$ N
B $261$ N
C $196$ N
D $392$ N
(c)
A particle of mass $10$ kg is now attached to the beam at $B$ . Find the new tension in the rope.
[2 marks]
Total for question 5: 4 marks
6
A uniform semicircular lamina has radius $6$ cm and diameter $AB$ .
(a)
What is the distance of the centre of mass of the lamina from $AB$ ?
[1 mark]
A $3.82$ cm
B $2.00$ cm
C $2.55$ cm
D $3.00$ cm
(b)
The lamina is freely suspended from $A$ . What angle does $AB$ make with the vertical?
[1 mark]
A $67.0^\circ$
B $23.0^\circ$
C $18.4^\circ$
D $32.5^\circ$
(c)
The lamina has mass $0.9$ kg. A particle of mass $0.3$ kg is attached to the lamina at $A$ . Find the distance of the centre of mass of the combined body from $AB$ .
[2 marks]
Total for question 6: 4 marks
7
A uniform wire is bent to form a right-angled triangular frame $PQR$ , with $PQ=8$ cm, $QR=6$ cm and $PR=10$ cm, and the right angle at $Q$ .
(a)
Find the distances of the centre of mass of the frame from $PQ$ and from $QR$ .
[3 marks]
(b)
The frame is freely suspended from $P$ . Find the angle between $PQ$ and the vertical.
[4 marks]
Total for question 7: 7 marks
8
A uniform lamina is in the shape of the letter T. It is made from a horizontal bar $20$ cm long and $4$ cm wide, and a vertical stem $4$ cm wide and $12$ cm long, attached below the middle of the bar. The top edge of the bar is $AB$ , with $A$ at the left-hand end, and $M$ is the midpoint of $AB$ .
(a)
Find the distance of the centre of mass of the lamina from $AB$ . The lamina is freely suspended from $A$ . Find the angle between $AB$ and the vertical.
[6 marks]
(b)
The lamina has mass $3.2$ kg. A particle of mass $m$ kg is attached to the lamina at $B$ . The combined body is freely suspended from $M$ and rests in equilibrium with its axis of symmetry at an angle of $30^\circ$ to the vertical. Find $m$ .
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).