2 Fig. 2.1 shows the positions of the points \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S } , \mathrm { T } , \mathrm { U } , \mathrm { V }\) and W which are at the vertices of a cube of side \(a\); Fig. 2.1 also shows coordinate axes, where O is the mid-point of PQ .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_510_494_365_788}
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\caption{Fig. 2.1}
\end{figure}
An open box, A, is made from thin uniform material in the form of the faces of the cube with just the face TUVW missing.
- Find the \(z\)-coordinate of the centre of mass of A .
Strips made of a thin heavy material are now fixed to the edges TW, WV and VU of box A, as shown in Fig. 2.2. Each of these three strips has the same mass as one face of the box. This new object is B.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_488_476_1388_797}
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
\end{figure} - Find the \(x\)-and \(z\)-coordinates of the centre of mass of B and show that the \(y\)-coordinate is \(\frac { 9 a } { 16 }\).
Object B is now placed on a plane which is inclined at \(\theta\) to the horizontal. B is positioned so that face PQRS is on the plane with SR at right angles to a line of greatest slope of the plane and with PQ higher than SR , as shown in Fig. 2.3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_237_284_2087_1555}
\captionsetup{labelformat=empty}
\caption{Fig. 2.3}
\end{figure} - Assuming that B does not slip, find \(\theta\) if B is on the point of tipping.
B is now placed on a different plane which is inclined at \(30 ^ { \circ }\) to the horizontal. When B is released it accelerates down the plane at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
- Calculate the coefficient of friction between B and the inclined plane.