6.04a Centre of mass: gravitational effect

2 A thin rigid rod PQ has length \(2 a\). Its mass per unit length, \(\rho\), is given by \(\rho = k \left( 1 + \frac { x } { 2 a } \right)\) where \(x\) is the distance from P and \(k\) is a positive constant. The mass of the rod is \(M\) and the moment of inertia of the rod about an axis through P perpendicular to PQ is \(I\).

1. Show that \(I = \frac { 14 } { 9 } M a ^ { 2 }\). The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where \(x = \frac { 3 } { 2 } a\). The magnitude of the impulse of this blow is \(J\).
2. Find, in terms of \(a , J\) and \(M\), the initial angular speed of the rod.
3. Find, in terms of \(a , g\) and \(M\), the set of values of \(J\) for which the rod makes complete revolutions.

4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.
(9)

6. \begin{figure}[h] \end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.

1. Show that the moment of inertia of the rod about the edge of the table is \(\frac { 7 } { 3 } m a ^ { 2 }\). The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle \(\theta\), its angular speed is \(\dot { \theta }\). Assuming that the rod has not started to slip,
2. show that \(\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }\),
3. find the angular acceleration of the rod,
4. find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is \(\mu\).
5. Show that the rod starts to slip when \(\tan \theta = \frac { 4 } { 13 } \mu\)

5 Fig. 5 shows the curve with equation \(y = - x ^ { 2 } + 4 x + 2\).
The curve intersects the \(x\)-axis at P and Q . The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 4\) is occupied by a uniform lamina L . The horizontal base of L is OA , where A is the point \(( 4,0 )\). \begin{figure}[h] \end{figure}

1. 1. Explain why the centre of mass of L lies on the line \(x = 2\).
   2. In this question you must show detailed reasoning. Find the \(y\)-coordinate of the centre of mass of \(L\).
2. L is freely suspended from A . Find the angle AO makes with the vertical. The region bounded by the curve and the \(x\)-axis is now occupied by a uniform lamina M . The horizontal base of M is PQ.
3. Explain how the position of the centre of mass of M differs from the position of the centre of mass of \(L\).

3 In this question you must show detailed reasoning. [In this question you may use the formula: Volume of cone \(= \frac { 1 } { 3 } \times\) base area × height.]
The region between the line \(\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }\), where \(a > 0\), the \(x\)-axis and the \(y\)-axis is rotated about the \(y\)-axis to form a uniform right circular cone C with base radius \(a\).

1. Show that the centre of mass of C is \(\frac { 3 } { 4 } a\) from its base. The cone C is fixed on top of a uniform cube, B , of edge length \(2 a\), so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} The centre of mass of the combined shape lies on the boundary of C and B .
   The density of \(B\) is not equal to the density of \(C\).
2. Determine the exact value of \(\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }\).
   [0pt] [3]

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).

1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]

The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).

1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]

**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]

Answer only one of the following two alternatives. EITHER \includegraphics{figure_10a} An object is formed by attaching a thin uniform rod \(PQ\) to a uniform rectangular lamina \(ABCD\). The lamina has mass \(m\), and \(AB = DC = 6a\), \(BC = AD = 3a\). The rod has mass \(M\) and length \(3a\). The end \(P\) of the rod is attached to the mid-point of \(AB\). The rod is perpendicular to \(AB\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l_1\), through \(Q\) and perpendicular to the plane of the lamina, is \(3(8m + M)a^2\). [4] Show that the moment of inertia of the object about a smooth horizontal axis \(l_2\), through the mid-point of \(PQ\) and perpendicular to the plane of the lamina, is \(\frac{3}{4}(17m + M)a^2\). [2] Find expressions for the periods of small oscillations of the object about the axes \(l_1\) and \(l_2\), and verify that these periods are equal when \(m = M\). [8] OR A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x\) kg. The data are summarised as follows. $$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y\) kg. The data are summarised as follows. $$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$ Test, at the 5% significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. [9] A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\)'s Crown plants. Test, at the 5% significance level, whether Farmer \(A\)'s claim is justified. [5]

\includegraphics{figure_3} A uniform disc, of radius \(a\) and mass \(2M\), is attached to a thin uniform rod \(AB\) of length \(6a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).

1. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc. [4]

The object is free to rotate about the axis \(l\). The object is held with \(AB\) horizontal and is released from rest. When \(AB\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac{3}{5}\), the angular speed of the object is \(\sqrt{\left(\frac{2g}{5a}\right)}\).

1. Find the possible values of \(x\). [5]

A uniform solid cylinder has radius 0.7 m and height \(h\) m. A uniform solid cone has base radius 0.7 m and height 2.4 m. The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(θ°\), is increased gradually until the cone is about to topple.

1. Find the value of \(θ\) at which the cone is about to topple. [2]
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). [2]

The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics{figure_4}

1. Given that the solid immediately topples, find the least possible value of \(h\). [5]

\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]

\includegraphics{figure_6} A uniform lamina \(OABCD\) consists of a semicircle \(BCD\) with centre \(O\) and radius \(0.6\) m and an isosceles triangle \(OAB\), joined along \(OB\) (see diagram). The triangle has area \(0.36\) m\(^2\) and \(AB = AO\).

1. Show that the centre of mass of the lamina lies on \(OB\). [4]
2. Calculate the distance of the centre of mass of the lamina from \(O\). [4]

\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,

1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
2. State what is implied by the modelling assumption that the beam is uniform. [1]

David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,

1. find the distance of the centre of mass of the beam from \(C\). [4]

\includegraphics{figure_2} A uniform plank \(AB\) has weight 120 N and length 3 m. The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1\) m and \(CD = x\) m, as shown in Figure 2. The reaction of the support on the plank at \(D\) has magnitude 80 N. Modelling the plank as a rod,

1. show that \(x = 0.75\) [3]

A rock is now placed at \(B\) and the plank is on the point of tilting about \(D\). Modelling the rock as a particle, find

1. the weight of the rock, [4]
2. the magnitude of the reaction of the support on the plank at \(D\). [2]
3. State how you have used the model of the rock as a particle. [1]

\includegraphics{figure_2} Figure 2 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 8a\) and \(AE = 6a\). The point \(X\) is the mid-point of \(BD\) and \(XC = 4a\). The centre of mass of the lamina is at \(G\).

1. Show that \(GX = \frac{14}{15}a\). [6]

The mass of the lamina is \(M\). A particle of mass \(\lambda M\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.

1. Find the value of \(\lambda\). [3]

\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]

\includegraphics{figure_1} Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm, from a uniform circular lamina, of centre \(O\) and radius 24 cm. The point \(X\) lies on the diameter \(AOB\) of the lamina and \(AX = 16\) cm. The centre of mass of \(T\) is at the point \(G\).

1. Find \(AG\). [6]

The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(OB\). A small stud of mass \(\frac{1}{4}m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(AB\) horizontal. Modelling the stud as a particle,

1. find the mass of \(T\) in terms of \(m\). [4]

\includegraphics{figure_1} A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(ABC\), with \(\angle ABC = 90°\), \(AB = 12\) cm and \(BC = 21\) cm. The point \(O\) is 5 cm from \(AB\) and 5 cm from \(BC\), as shown in Figure 1.

1. Find the distance of the centre of mass of \(S\) from
   1. \(AB\),
   2. \(BC\). [9]

The set square is freely suspended from \(C\) and hangs in equilibrium.

1. Find, to the nearest degree, the angle between \(CB\) and the vertical. [3]

A uniform rectangular lamina \(ABCD\) has mass 20 kg and sides of lengths \(AB = 0.6\) m and \(BC = 1.8\) m. It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(AB\).

1. Show that the moment of inertia of the lamina about the axis is 22.2 kg m\(^2\). [3]

\includegraphics{figure_5} The lamina is released from rest with \(BC\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 N m about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).

1. Show that the angular acceleration is zero when \(\cos \theta = 0.25\). [3]
2. Hence find the maximum angular speed of the lamina. [5]

\(ABCD\) is a uniform lamina in the shape of a kite with \(BA = BC = 0.37\) m, \(DA = DC = 0.91\) m and \(AC = 0.7\) m (see diagram). The centre of mass of \(ABCD\) is \(G\). \includegraphics{figure_4}

1. Explain why \(G\) lies on \(BD\). [1]
2. Show that the distance of \(G\) from \(B\) is \(0.36\) m. [4]

The lamina \(ABCD\) is freely suspended from the point \(A\).

1. Determine the acute angle that \(CD\) makes with the horizontal, stating which of \(C\) or \(D\) is higher. [4]