L = (g2-g1) /r is equation for find out length for vertical, so in this equation ‘r’ is notified by______

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Option A: Rate of change of grade

Explanation & Concepts:
In surveying and highway engineering, vertical curves are provided to connect two intersecting gradients smoothly. The equation $L = \frac{g_2 - g_1}{r}$ determines the length of a vertical curve, where $r$ denotes the rate of change of grade per unit length or per chain.

Working Principle: Vertical curves (usually parabolic) ensure a gradual transition in elevation to maintain comfortable riding conditions and adequate sight distance. The length of the curve ($L$) is directly proportional to the total change in grade ($N = g_2 - g_1$) and inversely proportional to the maximum permissible rate of change of grade ($r$).

Key Formulas: $L = \frac{g_2 - g_1}{r} = \frac{N}{r}$ — Length of vertical curve, where $N$ is the total change of grade and $r$ is the rate of change of grade; $N = |g_1 - g_2|$ — Total deviation or change in grade

Key Points: A rate of change of grade ($r$) typically ranges between 0.05% to 0.1% per 20 m or 30 m chain for summit curves.; Parabolic curves are preferred for vertical alignment because they provide a constant rate of change of grade throughout the length of the curve.

Advantages: Ensures smooth transition between two different gradients.; Improves driver comfort and safety by preventing sudden vertical accelerations.

Disadvantages: Requires larger land area and earthwork volumes to achieve smooth vertical transitions.; Designing for adequate sight distance at low values of $r$ can significantly increase construction cost.

Applications: Design of summit and valley vertical curves in highway engineering.; Railway track gradient transition design.

Additional Info: Option B represents the numerator $(g_2 - g_1)$, which is the algebraic difference or total change of grade ($N$).. Option C refers to the unit length (such as 20 m or 30 m) used to measure distances along the survey line, not 'r'.

Answer: A is correct — The variable 'r' in the equation represents the rate of change of grade per unit distance.

Exam Tip: For competitive exams like GATE and SSC JE, remember that parabolic vertical curves maintain a constant rate of change of grade ($r = \frac{d^2y}{dx^2}$), which makes calculation of elevations straightforward.

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Option B: total change of grade

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Option C: Chain length

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Option D: All of the above

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