Particles at coordinate positions

\includegraphics{figure_1} Figure 1 shows a triangular lamina \(ABC\). The coordinates of \(A\), \(B\) and \(C\) are \((0, 4)\), \((9, 0)\) and \((0, -4)\) respectively. Particles of mass \(4m\), \(6m\) and \(2m\) are attached at \(A\), \(B\) and \(C\) respectively.

1. Calculate the coordinates of the centre of mass of the three particles, without the lamina. [4]

The lamina \(ABC\) is uniform and of mass \(km\). The centre of mass of the combined system consisting of the three particles and the lamina has coordinates \((4, \lambda)\).

1. Show that \(k = 6\). [3]
2. Calculate the value of \(\lambda\). [2]

The combined system is freely suspended from \(O\) and hangs at rest.

1. Calculate, in degrees to one decimal place, the angle between \(AC\) and the vertical. [3]

Three particles of masses 2 kg, 3 kg and \(m\) kg are positioned at the points with coordinates \((a, 3)\), \((3, -1)\) and \((-2, 4)\) respectively. Given that the centre of mass of the particles is at the point with coordinates \((0, 2)\), find

1. the value of \(m\), [4]
2. the value of \(a\). [4]

The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2x - x^2\) and the \(x\)-axis.

1. Show that the area of \(R\) is \(\frac{4}{3}\). [4]
2. Find the coordinates of the centre of mass of the lamina. [5]

A non-uniform plank \(AB\) of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\), the magnitude of the reaction at \(A\) is 35g N.

1. Suggest a suitable model for
   1. the plank, [2 marks]
   2. the woman.
2. Calculate the magnitude of the reaction at \(B\), giving your answer in terms of \(g\). [2 marks]
3. Explain briefly, in the context of the problem, the term 'non-uniform'. [2 marks]
4. Find the distance of the centre of mass of the plank from \(A\). [4 marks]

\includegraphics{figure_1} Figure 1 shows a plank \(AB\) of mass 40 kg and length 6 m, which rests on supports at each of its ends. The plank is wedge-shaped, being thicker at end \(A\) than at end \(B\). A woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\).

1. Suggest suitable modelling assumptions which can be made about
   1. the plank,
   2. the woman. [3 marks]
   Given that the reactions at each support are of equal magnitude,
2. find the magnitude of the reaction on the support at \(A\), [2 marks]
3. calculate the distance of the centre of mass of the plank from \(A\). [4 marks]

Four tools are attached to a board. The board is to be modelled as a uniform lamina and the four tools as four particles. The diagram shows the lamina, the four particles \(A\), \(B\), \(C\) and \(D\), and the \(x\) and \(y\) axes. \includegraphics{figure_3} The lamina has mass 5 kg and its centre of mass is at the point \((7, 6)\). Particle \(A\) has mass 4 kg and is at the point \((11, 2)\). Particle \(B\) has mass 3 kg and is at the point \((3, 6)\). Particle \(C\) has mass 7 kg and is at the point \((5, 9)\). Particle \(D\) has mass 1 kg and is at the point \((1, 4)\). Find the coordinates of the centre of mass of the system of board and tools. [5 marks]

A straight rod \(AB\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k(a + 2x)\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod. [5]

The region bounded by the curve \(y = 2e^{\frac{1}{2}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), is occupied by a uniform lamina.

1. Find the exact value of the \(y\)-coordinate of the centre of mass of the lamina. [6]

As shown in the diagram below, a uniform lamina occupies the closed region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = f(x)\) where $$f(x) = \begin{cases} 2e^{\frac{1}{2}x} & 0 \leq x \leq 2, \\ \frac{2}{3}(5-x)e & 2 \leq x \leq 5. \end{cases}$$ \includegraphics{figure_4}

1. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina. [7]

\includegraphics{figure_4} A uniform rod AB has mass 3 kg and length 4 m. The end A of the rod is in contact with rough horizontal ground. The rod rests in equilibrium on a smooth horizontal peg 1.5 m above the ground, such that the rod is inclined at an angle of \(25°\) to the ground (see diagram). The rod is in a vertical plane perpendicular to the peg.

1. Determine the magnitude of the normal contact force between the peg and the rod. [3]
2. Determine the range of possible values of the coefficient of friction between the rod and the ground. [5]