Standard +0.3 This is a standard mechanics problem involving forces on a slope with friction and a string at an angle. Part (i) requires resolving forces perpendicular and parallel to the slope with given coefficient of friction - straightforward application of equilibrium equations. Part (ii) involves finding deceleration using F=ma and then using kinematics (v=u+at). The 'show that' structure and nice numbers (coefficient √3/23 giving clean answer 23√3g) indicate this is a routine textbook-style question, slightly above average difficulty only due to the angled rope requiring careful resolution.
6. A ski slope is modelled as a rough slope at an angle of \(30 ^ { \circ }\) to the horizontal. A skier of mass 72 kg is being towed up the slope at a constant speed of \(7 \mathrm {~ms} ^ { - 1 }\) by a rope inclined at an angle of \(30 ^ { \circ }\) to the slope. The skier is modelled as a particle \(P\) and the coefficient of friction between the skier and the slope is \(\frac { \sqrt { 3 } } { 23 }\). This situation is represented in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-13_396_625_388_790}
i. Show that the value of the normal reaction between the skier and the slope is \(23 \sqrt { 3 } g\) and find a similar expression in terms of \(g\) for the exact value of the tension in the rope.
ii. The skier lets go of the tow rope. Find the time the skier travels for before coming instantaneously to rest, giving your answer as a rational number of seconds. [0pt]
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6. A ski slope is modelled as a rough slope at an angle of $30 ^ { \circ }$ to the horizontal. A skier of mass 72 kg is being towed up the slope at a constant speed of $7 \mathrm {~ms} ^ { - 1 }$ by a rope inclined at an angle of $30 ^ { \circ }$ to the slope. The skier is modelled as a particle $P$ and the coefficient of friction between the skier and the slope is $\frac { \sqrt { 3 } } { 23 }$. This situation is represented in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-13_396_625_388_790}\\
i. Show that the value of the normal reaction between the skier and the slope is $23 \sqrt { 3 } g$ and find a similar expression in terms of $g$ for the exact value of the tension in the rope.\\
ii. The skier lets go of the tow rope. Find the time the skier travels for before coming instantaneously to rest, giving your answer as a rational number of seconds.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Mechanics 2021 Q6 [14]}}