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    1. A uniform plank AB, of mass 3m and length h, leans against a uniform rectangular block of mass m, width a and height h. The plank lies in a vertical plane which is perpendicular to a vertical face of the block, and which contains the centre of mass of the block. The block and the end B of the plank are on a horizontal surface which is rough enough to prevent sliding, and the surface of the block in contact with A is smooth. The system is in equilibrium and the plank makes an angle θ with the horizontal.

    (i) Given that the block is on the point of toppling over, express cos θ in terms of a and h.

    2. Three uniform rods AB, AC, BC, have lengths 2.0m, 2.0m, 2.4m and weights 480N, 560N, 400N, respectively. The rods are freely jointed at A, B, C, and hang in equilibrium in a vertical plane with BC horizontal, supported by vertical wires attached to the points P on AB and Q on AC, where AP=AQ=0.5m. The tensions in the wires at P and Q are T1 and T2 respectively.

    (i) Show that T1=640N.
    (ii) Find the magnitude and the direction of the force acting on BC at B.

    3. A pair of steps can be modelled as a uniform rod AB, mass 24kg and length 2m, freely hinged at A to a uniform rod AC, of mass 6kg and length 2m. The mid-points of the rods are joined by a light inextensible string. The rods rest in a vertical plane on smooth horizontal ground, with each rod inclined to the horizontal at an angle θ, where tan θ = 5/4. Find the tension in the string and the horizontal and vertical components of the force acting on AB at A.
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    Taking moments (plank) about B:
    R h sinθ = 3mg h/2 cosθ.

    taking moments about the far bottom corner of the block (the corner that pivots the toppling):

    R h sinθ = mg a/2

    => 3mg h/2 cosθ = mg a/2

    => cosθ = a/(3h)
 
 
 

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