Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Balance the upslope horizontal-force component against downslope weight.
Worked solution
(a)
Balance the upslope horizontal-force component against downslope weight.
Along the plane, \(\displaystyle P \cos \alpha =Mg \sin \alpha\), giving \(\displaystyle P=\)\(30\) N.
Answer: \(30\) N
(b)
Include the inward component of the horizontal push as well as the weight.
Perpendicular to the plane, \(\displaystyle R=Mg \cos \alpha +P \sin \alpha =\)\(50\) N.
Answer: \(50\) N
(c)
Use the angle complementary to the normal direction correctly.
The inward component is \(\displaystyle P \sin \alpha =\)\(18\) N.
Answer: \(18\) 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) · 2 marks
- 1 mark: Balance the upslope horizontal-force component against downslope weight.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A horizontal force can change the normal reaction on an incline.
(b) · 2 marks
- 1 mark: Include the inward component of the horizontal push as well as the weight.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The reaction is perpendicular to the surface, not necessarily vertical.
(c) · 2 marks
- 1 mark: Use the angle complementary to the normal direction correctly.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A horizontal force can change the normal reaction on an incline.
Total: 6 marks.