Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Combine the two along-motion equations, including plane friction and the downslope weight component.
Worked solution
(a)
Combine the two along-motion equations, including plane friction and the downslope weight component.
The net external driving force is \(\displaystyle \frac{856}{13}\) N on total mass 16 kg, giving \(\displaystyle \frac{107}{26}\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(\displaystyle \frac{107}{26}\) \(\displaystyle \frac{m}{s^{2}}\)
(b)
Use the hanging particle’s equation with its own mass.
The descending hanging particle gives \(\displaystyle T=H(g-a)=\)\(\displaystyle \frac{765}{13}\) N.
Answer: \(\displaystyle \frac{765}{13}\) N
(c)
Resolve perpendicular to the plane for the contacting particle.
There is no normal acceleration and no normal string component, so \(\displaystyle R=Mg \cos \alpha =\)\(\displaystyle \frac{720}{13}\) N.
Answer: \(\displaystyle \frac{720}{13}\) 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: Combine the two along-motion equations, including plane friction and the downslope weight component.
- 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: Put friction on the plane particle only.
(b) · 2 marks
- 1 mark: Use the hanging particle’s equation with its own mass.
- 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 normal reaction uses its own mass, not the hanging mass or their sum.
(c) · 2 marks
- 1 mark: Resolve perpendicular to the plane for the contacting particle.
- 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: Put friction on the plane particle only.
Total: 6 marks.