Mathematics 9709 · AS & A Level · Representation of data
Representation of data — practice question
A uniform body is obtained by boring a cylindrical hole right through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section $ABCD$ that passes through the centre of mass of the body. $AB = CD = 0.7\,\text{m}$, $BC = AD = 0.4\,\text{m}$, and the centre of the hole is $0.1\,\text{m}$ from $AB$ and $0.2\,\text{m}$ from $AD$ (see diagram). The hole has cross-sectional area $0.03\,\text{m}^2$.
(i)[4]
Show that the centre of mass of the object is $0.212\,\text{m}$ from $AB$, and calculate its distance from $AD$.
(ii)[2]
The object weighs $70\,\text{N}$ and is resting on a rough horizontal surface, with $AD$ touching the surface. A vertically upward force of magnitude $F\,\text{N}$ acts at $C$. The object is on the point of toppling. Find the value of $F$.
(iii)[2]
The force acting at $C$ is removed, and the object is set on a rough plane inclined at an angle $\theta^\circ$ to the horizontal. $AD$ lies along a line of greatest slope, with $A$ above $D$. The plane is rough enough to stop sliding, and the object does not topple. Find the greatest possible value of $\theta$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Taking moments gives $(0.7\times0.4)\times0.2=(0.28-0.03)x+0.03\times0.1$” …