Difficulty: Medium
Correct Answer: directly proportional
Explanation:
Introduction:When a container of liquid undergoes linear acceleration, the free surface tilts until it becomes perpendicular to the resultant of gravitational acceleration and the imposed acceleration. This question asks how the inclination depends on the acceleration magnitude.
Given Data / Assumptions:
Concept / Approach:
The free surface aligns perpendicular to the resultant acceleration vector formed by g and the applied acceleration a. The tilt angle theta measured from the horizontal satisfies tan(theta) = a_parallel / g if a acts along the tank axis. Hence theta increases as a increases; for small angles, theta ≈ a / g, showing direct proportionality.
Step-by-Step Solution:
Step 1: Represent accelerations: gravity g downward and tank acceleration a along the incline.Step 2: The resultant acceleration magnitude is sqrt(g^2 + a^2), and the free surface is perpendicular to this vector.Step 3: The inclination angle theta relative to horizontal satisfies tan(theta) = a / g for the along-slope component.Step 4: Therefore, theta increases with a and is approximately proportional to a for small angles, matching “directly proportional”.Verification / Alternative check:
Plotting theta vs a from tan(theta) = a / g shows monotonic increase; for a much less than g, theta ≈ a / g in radians, confirming proportionality.
Why Other Options Are Wrong:
Equal to the acceleration: Dimensionally meaningless; angle is not in acceleration units.Inversely proportional: Contradicts tan(theta) = a / g.Independent or proportional to acceleration squared: Both inconsistent with the tan relation.
Common Pitfalls:
Forgetting that the free surface is always perpendicular to the resultant acceleration and misinterpreting “inclination” as a length rather than an angle.
Final Answer:
directly proportional
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