Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Use equilibrium to find actual friction before checking its maximum.
Worked solution
(a)
Perpendicular balance gives \(\displaystyle R=Mg \cos \alpha\). Along the plane, friction must balance Mg sin α. This calculates the requirement without yet assuming it is below the available cap.
Answer: \(72\) N normal; \(54\) N friction uphill.
(b)
The available cap is \(\displaystyle \mu R=\)\(72\) N. It exceeds the required \(54\) N, so static equilibrium is consistent and is not limiting.
Answer: \(72\) N cap; the particle stays at rest.
(c)
At impending upward motion friction acts down the plane. Along-plane equilibrium gives \(\displaystyle P=Mg \sin \alpha +\mu Mg \cos \alpha =\)\(126\) N. The parallel applied force leaves the normal reaction unchanged.
Answer: \(126\) N.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 3 marks
- 1 mark: Resolve weight perpendicular to the plane to find the reaction.
- 1 mark: Use along-plane equilibrium to determine the required friction and its direction.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A stationary rough-plane particle need not be in limiting equilibrium.
(b) · 3 marks
- 1 mark: Calculate μR from the normal reaction.
- 1 mark: Compare the requirement with the cap and interpret the strict inequality.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Test the required friction before assuming that release causes sliding.
(c) · 3 marks
- 1 mark: Choose downhill friction for impending upward motion.
- 1 mark: Use the unchanged normal reaction and the limiting equality in the along-plane balance.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A stationary rough-plane particle need not be in limiting equilibrium.
Total: 9 marks.