Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Apply Newton’s second law to both carts together, excluding their internal rod forces.
Worked solution
(a)
Apply Newton’s second law to both carts together, excluding their internal rod forces.
The external opposing total is 21 N on mass 7 kg, giving \(\displaystyle \frac{3m}{s^{2}}\).
Answer: \(3\) \(\displaystyle \frac{m}{s^{2}}\)
(b)
The rear cart needs backward resultant 9 N. Its separate resistance supplies only 3 N, so the rod must push it backwards by 6 N. A rod pushes the carts apart in this arrangement, so it is compressed.
Answer: 6 N, compression.
(c)
Distinguish the resultant on the front cart from its applied brake force.
The front-cart resultant is its mass times deceleration: 12 N. It also equals brake force minus the forward rod force.
Answer: \(12\) 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: Apply Newton’s second law to both carts together, excluding their internal rod forces.
- 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 negative assumed tension means the rod is in compression; a rope would instead slacken.
(b) · 3 marks
- 1 mark: Isolate the rear cart and include its separate resistance.
- 1 mark: Use the common acceleration to determine the internal force and classify its push as compression.
- 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 count the internal rod force in the whole-system resultant.
(c) · 2 marks
- 1 mark: Distinguish the resultant on the front cart from its applied brake 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: A negative assumed tension means the rod is in compression; a rope would instead slacken.
Total: 7 marks.