Particles at coordinate positions

1. Three particles of masses \(3 m , m\) and \(2 m\) are positioned at the points with coordinates \(( a , 8 ) , ( - 4,0 )\) and \(( 5 , - 2 )\) respectively.

Given that the centre of mass of the three particles is at the point with coordinates \(( k , 2 k )\), where \(k\) is a constant, find the value of \(a\).
(5)

1. Three particles of masses \(m , 4 m\) and \(k m\) are placed at the points whose coordinates are \(( - 3,2 ) , ( 4,3 )\) and \(( 6 , - 4 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( c , 0 )\).

Find

1. the value of \(k\),
2. the value of \(c\).

1. Three particles of mass \(3 m , 2 m\) and \(k m\) are placed at the points whose coordinates are \(( 1,5 ) , ( 6,4 )\) and \(( a , 1 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( 3,3 )\).

Find

1. the value of \(k\),
2. the value of \(a\).

2 The diagram shows four particles, \(A , B , C\) and \(D\), which are fixed in a horizontal plane which contains the \(x\) - and \(y\)-axes, as shown. Particle \(A\) has mass 2 kg and is attached at the point ( 9,6 ).
Particle \(B\) has mass 3 kg and is attached at the point ( 2,4 ).
Particle \(C\) has mass 8 kg and is attached at the point \(( 3,8 )\).
Particle \(D\) has mass 7 kg and is attached at the point \(( 6,11 )\). \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-2_748_774_1402_625} Find the coordinates of the centre of mass of the four particles.

2 A uniform rod \(A B\), of mass 4 kg and length 6 metres, has three masses attached to it. A 3 kg mass is attached at the end \(A\) and a 5 kg mass is attached at the end \(B\). An 8 kg mass is attached at a point \(C\) on the rod. Find the distance \(A C\) if the centre of mass of the system is 4.3 m from point \(A\).
[0pt] [4 marks]

4 A light inextensible string of length 0.6 m has one end fixed to a point \(A\) on a smooth horizontal plane. The other end of the string is attached to a particle \(B\), of mass 0.4 kg , which rotates about \(A\) with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) on the surface of the plane.

1. Calculate the tension in the string. A particle \(P\) of mass 0.1 kg is attached to the mid-point of the string. The line \(A P B\) is straight and rotation continues at \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Calculate the tension in the section of the string \(A P\).
3. Calculate the total kinetic energy of the system.

4 A shovel consists of a blade and handle, as shown in Fig. 4.1 and Fig. 4.2. The dimensions shown in the figures are in metres.
The blade is modelled as a uniform rectangular lamina ABCD lying in the Oxy plane, where O is the mid-point of AB . The handle is modelled as a thin uniform rod EF . The handle lies in the Oyz plane, and makes an angle \(\alpha\) with \(\mathrm { O } y\), where \(\sin \alpha = \frac { 7 } { 25 }\). The rod and lamina are rigidly attached at E, the mid-point of CD.
The blade of the shovel has mass 1.25 kg and the handle of the shovel has mass 0.5 kg . \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Find,
   1. the \(y\)-coordinate of the centre of mass of the shovel,
   2. the \(z\)-coordinate of the centre of mass of the shovel. The shovel is freely suspended from O and hangs in equilibrium.
2. Calculate the angle that OE makes with the vertical.

5 In this question, all coordinates refer to the axes shown in Fig. 5.1. Fig. 5.1 shows a system of four particles with masses \(4 m , 3 m , m\) and \(2 m\) at the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . These points have coordinates \(( - 3,4 ) , ( 0,0 ) , ( 2,0 )\) and \(( 5,4 )\). \begin{figure}[h] \end{figure}

1. Calculate the coordinates of the centre of mass of the system of particles. A thin uniform rigid wire of mass \(12 m\) connects the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D with straight line sections, as shown in Fig. 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-5_460_903_1338_573} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
2. Calculate the coordinates of the centre of mass of the wire. The particles at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are now fixed to the wire to form a rigid object, \(R\).
3. Calculate the \(x\)-coordinate of the centre of mass of \(R\).

6 Fig. 6.1 shows a light rod ABC , of length 60 cm , where B is the midpoint of AC . Particles of masses \(3.5 \mathrm {~kg} , 1.4 \mathrm {~kg}\) and 2.1 kg are attached to \(\mathrm { A } , \mathrm { B }\) and C respectively. \begin{figure}[h] \end{figure} The centre of mass is located at a point G along the rod.

1. Determine the distance AG . Two light inextensible strings, each of length 40 cm , are attached to the rod, one at A , the other at C. The other ends of these strings are attached to a fixed point D. The rod is allowed to hang in equilibrium.
2. Determine the angle AD makes with the vertical. The two strings are now replaced by a single light inextensible string of length 80 cm . One end of the string is attached to A and the other end of the string is attached to C. The string passes over a smooth peg fixed at D. The rod hangs in equilibrium, but is not vertical, as shown in Fig. 6.2. Fig. 6.2
3. Explain why angle ADG and angle CDG must be equal.
4. Determine the tension in the string.

2 The diagram shows a system of three particles of masses \(3 m , 5 m\) and \(2 m\) situated in the \(x - y\) plane at the points \(\mathrm { A } ( 1,2 ) , \mathrm { B } ( 2 , - 2 )\) and \(\mathrm { C } ( 5,3 )\) respectively.
Determine the coordinates of the centre of mass of the system of particles.

3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A composite body consists of a uniform rod, \(A B\), and a particle.
The rod has length 4 metres and mass 22.5 kilograms.
The particle, \(P\), has mass 20 kilograms and is placed on the rod at a distance of 0.3 metres from \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-04_163_1323_767_402} 3

1. Find the distance of the centre of mass of the body from \(A\). 3
2. The body rests in equilibrium in a horizontal position on two supports, \(C\) and \(D\).
   The support \(C\) is 0.5 metres from \(A\) and the support \(D\) is 1 metre from \(B\). Find the magnitudes of the forces exerted on the body by the supports.
   [0pt] [4 marks]

1. Three particles of masses \(4 m , 2 m\) and \(k m\) are placed at the points with coordinates \(( - 3 , - 1 ) , ( 6,1 )\) and \(( - 1,5 )\) respectively.

Given that the centre of mass of the three particles is at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. show that \(\bar { x } = \frac { - k } { k + 6 }\)
2. find \(\bar { y }\) in terms of \(k\). Given that the centre of mass of the three particles lies on the line with equation \(y = 2 x + 3\)
3. find the value of \(k\). A fourth particle is placed at the point with coordinates \(( \lambda , 4 )\).
   Given that the centre of mass of the four particles also lies on the line with equation \(y = 2 x + 3\)
4. find the value of \(\lambda\).

1. Three particles of masses \(3 m\), \(4 m\) and \(2 m\) are placed at the points \(( - 2,2 ) , ( 3,1 )\) and ( \(p , p\) ) respectively.

The value of \(p\) is such that the distance of the centre of mass of the three particles from the point ( 0,0 ) is as small as possible. Find the value of \(p\).

1. Three particles of masses \(2 m , 3 m\) and \(k m\) are placed at the points with coordinates (3a, 2a), (a, -4a) and (-3a, 4a) respectively.

The centre of mass of the three particles lies at the point with coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Find \(\bar { x }\) in terms of \(a\) and \(k\)
   2. Find \(\bar { y }\) in terms of \(a\) and \(k\) Given that the distance of the centre of mass of the three particles from the point ( 0,0 ) is \(\frac { 1 } { 3 } a\)
2. find the possible values of \(k\)

1. Three particles of masses \(3 m , 4 m\) and \(k m\) are positioned at the points with coordinates ( \(2 a , 3 a\) ), ( \(a , 5 a\) ) and ( \(2 \mu a , \mu a\) ) respectively, where \(k\) and \(\mu\) are constants.

The centre of mass of the three particles is at the point with coordinates \(( 2 a , 4 a )\).
Find (i) the value of \(k\) (ii) the value of \(\mu\)

2 A piece of modern art is modelled as a uniform lamina and three particles. The diagram shows the lamina, the three particles \(A , B\) and \(C\), and the \(x\) - and \(y\)-axes. \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-2_875_1004_1414_502} The lamina, which is fixed in the \(x - y\) plane, has mass 10 kg and its centre of mass is at the point (12, 9). The three particles are attached to the lamina.
Particle \(A\) has mass 3 kg and is at the point (15, 6).
Particle \(B\) has mass 1 kg and is at the point ( 7,14 ).
Particle \(C\) has mass 6 kg and is at the point ( 8,7 ).
Find the coordinates of the centre of mass of the piece of modern art.

3 Three particles are attached to a light rectangular lamina \(O A B C\), which is fixed in a horizontal plane. Take \(O A\) and \(O C\) as the \(x\) - and \(y\)-axes, as shown. Particle \(P\) has mass 1 kg and is attached at the point \(( 25,10 )\).
Particle \(Q\) has mass 4 kg and is attached at the point ( 12,7 ).
Particle \(R\) has mass 5 kg and is attached at the point \(( 4,18 )\). \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-3_782_1033_703_482} Find the coordinates of the centre of mass of the three particles.

1 A rigid rod, \(A B\), has mass 2 kg and length 4 metres.
Two particles of masses 5 kg and 3 kg are fixed to \(A\) and \(B\) respectively to create a composite body, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-02_120_730_769_653} Find the distance of the centre of mass of the composite body from \(B\). Circle your answer.
1.5 metres
1.6 metres
2.4 metres
2.5 metres

3 Three particles are attached to a light rod, \(A B\), of length 40 cm The particles are attached at \(A , B\) and the midpoint of the rod.
The particle at \(A\) has mass 5 kg
The particle at \(B\) has mass 1 kg
The particle at the midpoint has mass 4 kg
Find the distance of the centre of mass of this system from the midpoint of the rod.
Circle your answer.
[0pt] [1 mark] \(4 \mathrm {~cm} \quad 8 \mathrm {~cm} \quad 12 \mathrm {~cm} \quad 28 \mathrm {~cm}\) Turn over for the next question

\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{3}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]
2. The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\). Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]

A child's toy consists of an iron disc of radius \(r\) and a vertical bead with \(3r\) at rail that is rigidly fixed to the disc so that the toy rocks as it rolls. The circumference of the disc is such that the disc and bead have the same material. Show that the centre of mass of the toy is at a distance \(\frac{27r}{10}\) from the centre of the disc. [4]

Three particles of mass \(3m\), \(5m\) and \(2m\) are placed at points with coordinates \((4, 0)\), \((0, -3)\) and \((4, 2)\) respectively. The centre of mass of the system of three particles is at \((2, k)\).

1. Show that \(λ = 2\). [4]

1. Calculate the value of \(k\). [3]

Three particles of mass \(3m\), \(5m\) and \(\lambda m\) are placed at points with coordinates \((4, 0)\), \((0, -3)\) and \((4, 2)\) respectively. The centre of mass of the system of three particles is at \((2, k)\).

1. Show that \(\lambda = 2\). [4]
2. Calculate the value of \(k\). [3]