Centre of mass of composite lamina

2 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_291_732_822_708} A uniform lamina \(A B C D E\) consists of a rectangular part with sides 5 cm and 10 cm , and a part in the form of a quarter of a circle of radius 5 cm , as shown in the diagram.

1. Show that the distance of the centre of mass of the part \(C D E\) of the lamina is \(\frac { 20 } { 3 \pi } \mathrm {~cm}\) from \(C E\).
2. Find the distance of the centre of mass of the lamina \(A B C D E\) from the edge \(A B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_597_690_1416_731} A large uniform lamina is in the shape of a right-angled triangle \(A B C\), with hypotenuse \(A C\), joined to a semicircle \(A D C\) with diameter \(A C\). The sides \(A B\) and \(B C\) have lengths 3 m and 4 m respectively, as shown in the diagram.

1. Show that the distance from \(A B\) of the centre of mass of the semicircular part \(A D C\) of the lamina is \(\left( 2 + \frac { 2 } { \pi } \right) \mathrm { m }\).
2. Show that the distance from \(A B\) of the centre of mass of the complete lamina is 2.14 m , correct to 3 significant figures.

1 A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x - y\) plane. The coordinates of the vertices are \(O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )\) and \(C ( 3,4 )\). Find the \(x\)-coordinate of the centre of mass of the lamina.

6 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-3_582_862_577_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).

1. Show that the centre of mass of the lamina lies on \(O B\).
2. Calculate the distance of the centre of mass of the lamina from \(O\).

2 The region bounded by the \(x\)-axis and the curve \(y = a x ( 2 - x )\), where \(a\) is a constant, is occupied by a uniform lamina \(L _ { 1 }\) (see Fig. 1). Units on the axes are metres. \begin{figure}[h] \end{figure}

1. Write down the value of the \(x\)-coordinate of the centre of mass of \(L _ { 1 }\).
2. Show that the \(y\)-coordinate of the centre of mass of \(L _ { 1 }\) is \(\frac { 2 } { 5 } a\). The mass of \(L _ { 1 }\) is \(M \mathrm {~kg}\). A uniform rectangular lamina of width 2 m and height \(a \mathrm {~m}\) is made from a different material from that of \(L _ { 1 }\) and has a mass of \(2 M \mathrm {~kg}\). A new lamina, \(L _ { 2 }\), is formed by joining the straight edge of \(L _ { 1 }\) to an edge of the rectangular lamina of length 2 m (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8c9d007-e67f-4637-9e74-630ba9a91442-2_547_273_1772_890} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(L _ { 2 }\) is freely suspended from one of its right-angled corners and hangs in equilibrium with its edge of length 2 m making an angle of \(20 ^ { \circ }\) with the horizontal.
3. Find the value of \(a\), giving your answer correct to 3 significant figures.

6 Nm
8 Nm
10 Nm
14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5

1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\) [0pt] [5 marks]
   5
2. The solid is suspended in equilibrium from a point on the edge of the circular face.
   Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
   6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
   The other end of the string is attached to a fixed point \(O\) The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
   1. Find the speed of the sphere when it is directly below \(O\) 6
   2. State one assumption that you made about the string.
      6
   3. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
   4. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
      Explain why this result is reasonable, showing any calculations that you make.

\includegraphics{figure_2} The uniform L-shaped lamina \(OABCDE\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from

1. \(OE\). [4]
2. \(OA\). [4]

The lamina is freely suspended from \(O\) and hangs in equilibrium with \(OE\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac{4}{3}\).

1. Find the value of \(a\). [4]