Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Use equilibrium and then check the static-friction inequality.
Worked solution
(a)
The applied pull is horizontal. Vertical equilibrium therefore gives \(\displaystyle R=Mg=40 N\).
Answer: 40 N.
(b)
Horizontal balance requires \(\displaystyle F=\)\(6\) N, while \(\displaystyle \mu R=\)\(12\) N. The required friction is strictly below the cap.
Answer: \(6\) N left; equilibrium is possible.
(c)
Combine the perpendicular actual friction and normal components by Pythagoras. The contact direction has tangent \(\displaystyle \frac{F}{R}=\)\(\displaystyle \frac{3}{20}\).
Answer: \(40.4\) N, 8.5° towards the left.
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: Identify and balance the vertical forces without adding the horizontal pull to that equation.
- 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: Use actual friction rather than μR when equilibrium is not limiting.
(b) · 3 marks
- 1 mark: Identify the required friction magnitude and its leftward direction from equilibrium.
- 1 mark: Calculate the limiting value and compare it with the required friction.
- 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: Distinguish the normal component from the entire contact-force magnitude.
(c) · 3 marks
- 1 mark: Use actual friction and normal reaction as perpendicular components.
- 1 mark: Find magnitude and the angle relative to the stated upward normal, using the requested precision.
- 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: Use actual friction rather than μR when equilibrium is not limiting.
Total: 8 marks.