Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Solve the horizontal limiting equality and normal balance simultaneously.
Worked solution
(a)
Use \(\displaystyle R=70-\frac{3}{5}P\) and \(\displaystyle \frac{4}{5}P=\frac{1}{5}R\). Elimination gives \(\displaystyle P= \frac{\mu Mg}{\cos \alpha + \mu \sin \alpha }\), hence the stated pull.
Answer: \(\displaystyle P=\frac{350}{23}\) N.
(b)
Substituting the limiting pull into \(\displaystyle R=Mg-P \sin \alpha\) gives \(\displaystyle \frac{1400}{23}\) N, which is positive.
Answer: \(\displaystyle \frac{1400}{23}\) N; contact is maintained.
(c)
Use the changed normal reaction in the limiting-friction formula.
The limit μR equals \(\displaystyle \frac{280}{23}\) N and balances P cos α.
Answer: \(\displaystyle \frac{280}{23}\) 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 vertically to express the reaction in terms of the unknown pull.
- 1 mark: Use \(\displaystyle F=\mu R\) and horizontal equilibrium to eliminate the reaction and solve for P.
- 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: Do not set \(\displaystyle R=Mg\) when the applied force has a vertical component.
(b) · 3 marks
- 1 mark: Evaluate the normal balance with the determined upward pull.
- 1 mark: Check the nonnegative-reaction condition for the contact model.
- 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: An upward pull and a downward push change the reaction in opposite directions.
(c) · 2 marks
- 1 mark: Use the changed normal reaction in the limiting-friction formula.
- 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: Do not set \(\displaystyle R=Mg\) when the applied force has a vertical component.
Total: 8 marks.