6 A block B of mass \(m \mathrm {~kg}\) rests on a rough slope inclined at angle \(\alpha\) to the horizontal. The coefficient of friction between \(B\) and the slope is \(\frac { 5 } { 9 }\).
- When B is in limiting equilibrium, show that \(\tan \alpha = \frac { 5 } { 9 }\).
- If \(\alpha = 40 ^ { \circ }\), determine the acceleration of B down the slope.
A horizontal force of magnitude \(P \mathrm {~N}\) is now applied to B , as shown in the diagram below. At first B is at rest.
\includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-7_381_410_689_242}
\(P\) is gradually increased. - Show that, for B to slide on the slope,
$$\mathrm { P } \left( \cos \alpha - \frac { 5 } { 9 } \sin \alpha \right) > \mathrm { mg } \left( \frac { 5 } { 9 } \cos \alpha + \sin \alpha \right) .$$
- Determine, in degrees, the least value of \(\alpha\) for which B will not slide no matter how large \(P\) becomes.