Physics • Year 11 • Module 2: Dynamics • Lesson 2
Vector Forces — Resolution and Equilibrium
Build HSC Band 5–6 extended-response technique: multi-step vector analysis, experimental design and evaluation of force equilibrium in real structural contexts.
1. Multi-step calculation — V8 Supercar pit crew analysis (Band 5–6)
8 marks Band 5–6
Scenario. During a V8 Supercar pit stop, two crew members push the stationary car simultaneously. Crew member A applies a force of 320 N at 25° north of east. Crew member B applies a force of 280 N at 40° south of east. A third force (friction from the floor) acts on the car.
Take east as the positive x-direction and north as the positive y-direction. g = 9.8 m s−2.
(a) Resolve the force of crew member A into x and y components. Show working and state the sign of each component. (2 marks)
(b) Resolve the force of crew member B into x and y components, taking care with signs. Show working. (2 marks)
(c) Calculate the magnitude and direction of the resultant of forces A and B (before friction). Use Pythagoras and inverse tangent. State the direction clearly as “X° north/south of east.” (2 marks)
(d) The car remains stationary (in equilibrium). Calculate the magnitude and direction of the friction force that achieves this. Explain why “friction acts backward” is not a complete description. (2 marks)
2. Experimental design — testing the equilibrium condition (Band 5–6)
7 marks Band 5–6
Research question. A Year 11 student claims: “For a mass hanging from two cables at equal angles, the tension formula T = W / (2 sin θ) is only accurate for angles greater than 30°.” Design a scientific investigation to test this claim across a range of angles.
Constraints: You have access to a standard Year 11 physics laboratory: two spring balances (0–20 N, ±0.1 N), string, a protractor (±1°), a set of known masses (100–500 g), a retort stand, and a ruler. Investigation must be completed in one 90-minute lesson.
Q2. Design the investigation and present it in the format below.
- State a testable hypothesis including the independent and dependent variables.
- Identify the independent variable, dependent variable, and at least two controlled variables.
- Describe the procedure in at least four numbered steps, including how you will measure T and θ precisely.
- Explain what result would falsify the student’s claim.
- State two limitations of your design and one improvement to increase reliability.
Q1 — Sample Band 6 response (8 marks), annotated
(a) Crew A components (2 marks):
Positive direction: east = +x, north = +y.
Ax = 320 cos 25° = 320 × 0.906 = 290.0 N (eastward, positive) [1].
Ay = 320 sin 25° = 320 × 0.423 = 135.2 N (northward, positive) [1].
(b) Crew B components (2 marks):
Bx = 280 cos 40° = 280 × 0.766 = 214.5 N (eastward, positive) [1].
By = −280 sin 40° = −280 × 0.643 = −180.0 N (southward, negative) [1].
Marking note: Deduct 1 mark if the sign on By is positive (direction error).
(c) Resultant of A + B (2 marks):
Total Fx = 290.0 + 214.5 = 504.5 N.
Total Fy = 135.2 + (−180.0) = −44.8 N.
Resultant magnitude = √(504.5² + 44.8²) = √(254,600 + 2,007) = √256,607 ≈ 506.6 N [1].
Direction: θ = tan−1(44.8 / 504.5) = tan−1(0.0888) = 5.1°. Since Fy is negative (south), the resultant is 5.1° south of east [1].
(d) Friction force for equilibrium (2 marks):
For ΣFx = 0 and ΣFy = 0, the friction force must be equal in magnitude to the resultant (506.6 N) and exactly opposite in direction: 506.6 N directed 5.1° north of west [1].
“Friction acts backward” is incomplete because it specifies only the x-component (west/backward) without addressing the north component. The friction force has a northward component (+44.8 N) in addition to its westward component (−504.5 N). A complete vector description requires both magnitude and direction [1].
Marking criteria summary: 1 = correct Ax calculation; 1 = correct Ay with positive sign; 1 = correct Bx calculation; 1 = correct By with negative sign; 1 = correct resultant magnitude using Pythagoras; 1 = correct resultant direction stated as “south of east” with angle; 1 = correct friction magnitude and direction; 1 = explains why “backward” is incomplete (missing northward component).
Q2 — Sample Band 6 response (7 marks), annotated
Hypothesis: The formula T = W / (2 sin θ) will predict the tension in each cable accurately (within ±5%) for all tested angles from 10° to 80° — not only for angles above 30°. Independent variable: cable angle θ (10°, 20°, 30°, 45°, 60°, 75°). Dependent variable: tension T measured by spring balance. Controlled variables: suspended mass (e.g. 500 g = 4.9 N), string length on each side, symmetric setup (equal angles both sides) [1].
Procedure: (1) Set up a horizontal rigid bar on a retort stand. Attach two lengths of string to the bar at a measured distance apart; bring them together at a central knot and hang a 500 g mass from the knot. (2) Adjust the angle of each string to the horizontal using the protractor, reading θ (±1°). Ensure both sides are symmetric. (3) Insert a calibrated spring balance into each string at a point near the bar; record the reading T from each balance (±0.1 N). (4) Repeat for six different angles: 10°, 20°, 30°, 45°, 60°, and 75°. For each angle, calculate the predicted Tpred = 4.9 / (2 sin θ) and compare it with the measured value Tmeas [1].
Falsification: The student’s claim would be falsified if the measured tension agrees with Tpred at angles of 10° and 20° as well as at 30° and above — showing the formula is accurate across all tested angles. The claim would be supported only if the percentage error |Tmeas − Tpred| / Tpred × 100% were systematically larger below 30° than above [1].
Limitations: (1) At very shallow angles (e.g. θ = 10°), the theoretical tension is ~707 N — far exceeding the 20 N spring balance range, making direct measurement at low angles impossible with the given equipment [1]. (2) Reading the angle θ with a hand-held protractor introduces ±1° error; at steep angles (small θ) this becomes a large fractional error in sin θ and therefore in T [1].
Improvement: Repeat each angle three times and average the spring balance readings to improve precision; alternatively, use a digital force sensor instead of a spring balance to extend the measurable range [1].
What the results should show: For angles of 20° and above (within the spring balance range), the measured T should match Tpred within ±5%, demonstrating the formula holds at all measurable angles. The student’s claim would not be supported by the data [1].
Marking criteria summary (7 marks): 1 = testable hypothesis with IV and DV; 1 = four clear procedure steps including how T and θ are measured; 1 = states what would falsify the claim (formula accurate at low angles too); 1 = valid limitation 1 (spring balance range inadequate at low angles); 1 = valid limitation 2 (angle measurement error); 1 = one specific improvement; 1 = precise physics terminology (equilibrium, component, ΣFy = 0, independent/dependent/controlled variable) throughout.