Equilibrium of rigid bodies

Model 2 is a simplified model for the sign. In this model, the lamina is in the shape of a trapezium with vertices at the same points O, A, B and C as in Model 1 (see diagram).

The centre of mass of the lamina in Model 2 is at the point with coordinates ( .

Find the value of and the value of

The sign is placed on a rough plane inclined at 50° to the horizontal. The line OA rests on the plane with O below A. The line OA lies along a line of greatest slope of the plane.

You may assume that all forces acting on the sign lie in the vertical plane containing OA and the plane is sufficiently rough to prevent sliding.

You are given that and from Model 1 are both within 0.04 m of the actual coordinates of the centre of mass of the sign.

Determine whether the sign will topple. Justify your answer.

Use your answer to part (b) to explain whether Model 2 would have predicted the same

result as in part (c)(i).

A shape, S, is formed by attaching a particle of mass 2 m kg to the vertex of a uniform solid cone of mass 8 m kg. The height of the cone is h m and the radius of the base of the cone is 1.1 m.

Explain why the centre of mass of S must lie on the central axis of the cone.

Two strings are attached to S, one at the vertex of the cone and one at A which is a point on the edge of the base of S. The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to S at A is inextensible and has length 1.6 m. The string attached to S at the vertex is elastic with modulus of elasticity 8 mg N.

Shape S is in equilibrium with its axis horizontal (see diagram).

Determine the natural length of the elastic string.

In this question you must show detailed reasoning.

The region bounded by the x‑axis, the y‑axis, the line x = 4 and the curve with equation

The centre of mass of the lamina is at the point G ) (see diagram).

Determine the value of . Give your answer correct to 3 significant figures.

P is the point on the curved edge of the lamina where x = 3. The lamina is freely suspended from P and hangs in equilibrium in a vertical plane.

Determine the acute angle that the longest straight edge of the lamina makes with the vertical.

Find giving your answer correct to 3 significant figures.

Using the modelling assumptions, determine the value of P giving your answer correct to

3 significant figures.

Fig. 7.1 shows a uniform lamina in the shape of a sector of a circle of radius r and angle 2 θ where θ is in radians. The sector consists of a triangle OAB and a segment bounded by the chord AB.

Explain why the centre of mass of the segment lies on the radius through the midpoint of AB.

A uniform circular lamina of radius 5 units is placed with its centre at the origin, O, of an x - y coordinate system. A component for a machine is made by removing and discarding a segment from the lamina. The radius of the circle from which the segment is formed is 3 units and the centre of this circle is O. The centre of the straight edge of the segment has coordinates (0, 2) and this edge is perpendicular to the y-axis (see Fig. 7.2).

Find the y-coordinate of the centre of mass of the component, giving your answer correct to 3

significant figures.

C is the point on the component with coordinates (0, 5). The component is now placed horizontally and supported only at O. A particle of mass m kg is placed on the component at C and the component and particle are in equilibrium.

Find the mass of the component in terms of m.