Two forces A and B act on a 19.7kg particle. The larger force in magnitude is A. When both forces point due east the magnitude of the acceleration of the mass is 0.585m/s2. When A points due east and B points due west the mass accelerates at 0.477m/s2 due east. What is the magnitude of force A?

Question

BlackTom AI · Accepted Answer

Question restatement: A 19.7 kg particle is acted on by two forces A and B. A has the greater magnitude. When both forces point due east, the acceleration is 0.585 m/s^2 east. When A points east and B points west, the acceleration is 0.477 m/s^2 east. What is the magnitude of force A?

Option-by-option analysis:
Option 1: none of these
This would be correct only if none of the numerical values matched the situation. Since we can set up equations from the given accelerations that lead to a specific numeric value for A (as shown below), this option is not consistent with the data. Therefore, this choice is unlikely.

Option 2: 12 N
If A were 12 N, we can test consistency with the two scenarios. Scenario 1 (both east): A + B = m a = 19.7 × 0.585 ≈ 11.5245 N. But A + B would then be at least 12 N, already exceeding 11.5245 N, which contradicts the first scenario. Hence 12 N cannot be the correct magnitude for A.

Option 3: 4.4 N
If A were 4.4 N, Scenario 2 (A east, B west) would give A − B = m a = 19.7 × 0.477 ≈ 9.3969 N, implying B ≈ A − 9.3969 ≈ 4.4 − 9.3969 ≈ −4.9969 N, i.e., B would be negative in magnitude, which is impossible since B is a force with a definite magnitude (and the problem states A has the larger magnitude). This inconsistency shows 4.4 N cannot be the magnitude of A.

Option 4: 31 N
If A were 31 N, consider Scenario 1 (both east): A + B = 11.5245 N, but A by itself would already exceed that total, leaving B negative, which is not physically plausible for a magnitude. Thus 31 N is not compatible with the given accelerations.

Option 5: 10.5 N
Let us test this value against both scenarios. Scenario 1: A + B = 11.5245 N, so B ≈ 11.5245 − 10.5 ≈ 1.0245 N. Scenario 2: A − B = 9.3969 N, so B ≈ A − 9.3969 ≈ 10.5 − 9.3969 ≈ 1.1031 N. The two estimates for B (1.0245 N and 1.1031 N) are close and within rounding tolerance, supporting consistency. This makes sense physically since B is smaller than A as stated, and the signs align with the directions given in each scenario.

Numerical solution from the physics: Let m = 19.7 kg. Scenario with both east: A + B = m × 0.585 ≈ 19.7 × 0.585 ≈ 11.5245 N. Scenario with A east, B west: A − B = m × 0.477 ≈ 19.7 × 0.477 ≈ 9.3969 N. Solving these two equations:
- Adding gives 2A ≈ 20.9214 → A ≈ 10.4607 N.
- Substituting back, B ≈ 11.5245 − A ≈ 1.0638 N.
Thus the magnitude of A is about 10.46 N, which rounds to 10.5 N in the provided options.

Summary of reasoning flow: start from the two acceleration conditions to write two linear equations in A and B, solve for A and B, check consistency with A > B, and compare to the listed options. The only choice that aligns with the derived A value and with the condition that A is the larger force is 10.5 N. The other options lead to contradictions either with the given totals in Scenario 1 or with the implied signs and magnitudes of B, or violate the stated dominance of A over B.

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