Maximum or minimum mass

4. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 6.5 m and mass 1.2 kg . The centre of mass of the rod is 3 m from \(A\). The rod rests on a horizontal step and overhangs the end of the step \(C\) by 1.5 m , as shown in Figure 2. The rod is perpendicular to the edge of the step.
A particle of mass 4 kg is placed on the rod at \(B\) and another particle, whose mass is \(M \mathrm {~kg}\), is placed on the rod at \(D\), where \(A D = 0.5 \mathrm {~m}\). The rod remains in equilibrium in a horizontal position.

1. Find the smallest possible value of \(M\). The particle at \(B\) and the particle at \(D\) are now removed.
   A new particle is placed on the rod at the point \(E\), where \(E B = 0.9 \mathrm {~m}\).
   The rod remains in equilibrium in a horizontal position but is on the point of tilting about \(C\).
2. Find the magnitude of the force acting on the rod at \(C\).

4. \begin{figure}[h] \end{figure} A diving board \(A B\) consists of a wooden plank of length 4 m and mass 30 kg . The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(A C = 0.6 \mathrm {~m}\), as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.

1. Find
   1. the magnitude of the force acting on the plank at \(A\),
   2. the magnitude of the force acting on the plank at \(C\). The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N .
2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\).
3. Explain how you have used the fact that the diver is modelled as a particle.
   

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4 A uniform plank is 10 m long and has mass 15 kg . It is placed on horizontal ground at the edge of a vertical river bank, so that 2 m of the plank is projecting over the edge, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-3_250_1285_1361_388}

1. A woman of mass 50 kg stands on the part of the plank which projects over the river. Find the greatest distance from the river bank at which she can safely stand.
2. The woman wishes to stand safely at the end of the plank which projects over the river. Find the minimum mass which she should place on the other end of the plank so that she can do this.
3. State how you have used the fact that the plank is uniform in your solution.
4. State one other modelling assumption which you have made.

3 A side view of a free-standing kitchen cupboard on a horizontal floor is shown in Fig. 3.1. The cupboard consists of: a base CE; vertical ends BC and DE; an overhanging horizontal top AD. The dimensions, in metres, of the cupboard are shown in the figure. The cupboard and contents have a weight of 340 N and centre of mass at G . \begin{figure}[h] \end{figure}

1. Calculate the magnitude of the vertical force required at A for the cupboard to be on the point of tipping in the cases where the force acts
   (A) downwards,
   (B) upwards. A force of magnitude \(Q \mathrm {~N}\) is now applied at A at an angle of \(\theta\) to AB , as shown in Fig. 3.2, where \(\cos \theta = \frac { 5 } { 13 } \left( \right.\) and \(\left. \sin \theta = \frac { 12 } { 13 } \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-4_303_1134_1619_504} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
2. By considering the vertical and horizontal components of the force at A , show that the clockwise moment of this force about E is \(\frac { 30 Q } { 13 } \mathrm { Nm }\). With the force of magnitude \(Q \mathrm {~N}\) acting as shown in Fig. 3.2, the cupboard is in equilibrium and is on the point of tipping but not on the point of sliding.
3. Show that \(Q = 221\) and that the coefficient of friction between the cupboard base and the floor must be greater than \(\frac { 5 } { 8 }\).

\includegraphics{figure_1} A uniform rod \(AB\) has length \(2a\) and mass \(M\). The rod is held in equilibrium in a horizontal position by two vertical light strings which are attached to the rod at \(C\) and \(D\), where \(AC = \frac{2}{5}a\) and \(DB = \frac{3}{5}a\), as shown in Figure 1. A particle \(P\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the largest possible mass of the particle \(P\) [3] Given that the mass of \(P\) is \(\frac{1}{2}M\)
2. Find, in terms of \(M\) and \(g\), the tension in the string that is attached to the rod at \(C\). [3]