Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Balance the vertical component of tension against weight.
Worked solution
(a)
Balance the vertical component of tension against weight.
Vertical balance \(\displaystyle T \sin \alpha =Mg\) gives \(\displaystyle T=\)\(\displaystyle \frac{100}{3}\) N.
Answer: \(\displaystyle \frac{100}{3}\) N
(b)
Balance the horizontal tension component against the applied force.
Horizontal balance gives \(\displaystyle P=T \cos \alpha =\)\(\displaystyle \frac{80}{3}\) N.
Answer: \(\displaystyle \frac{80}{3}\) N
(c)
Divide the horizontal balance by the nonzero tension.
From horizontal balance, \(\displaystyle \frac{P}{T}= \cos \alpha =\)\(\displaystyle \frac{4}{5}\).
Answer: \(\displaystyle \frac{4}{5}\)
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 vertical component of tension against 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 closing force points opposite the resultant of the other forces.
(b) · 2 marks
- 1 mark: Balance the horizontal tension component against the applied force.
- 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: Lami’s theorem is an optional alternative, not required prior knowledge here.
(c) · 2 marks
- 1 mark: Divide the horizontal balance by the nonzero tension.
- 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 closing force points opposite the resultant of the other forces.
Total: 6 marks.