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If r is the radius of curvature at any point of a transition curve and l is the distance from thebeginning of the transition curve to that point, then for ideal transition
- a)1μ r
- b)1 r2
- c)1 μ 1/r
- d)1 μ 1/r2
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If r is the radius of curvature at any point of a transition curve and...
Yes, the correct answer is option C: "1μ 1/r."
In an ideal transition curve, the radius of curvature at any point is inversely proportional to the degree of curvature at that point. This means that as the curvature of the curve increases, the radius of curvature decreases, and vice versa. This relationship can be expressed mathematically as:
1μ 1/r = k
where μ is the degree of curvature, r is the radius of curvature, and k is a constant.
Option A: "1μ r" is incorrect.
Option B: "1 r2" is incorrect.
Option D: "1 μ 1/r2" is incorrect.
Most Upvoted Answer
If r is the radius of curvature at any point of a transition curve and...
Explanation:
Transition curve is a curve provided to connect a straight line and a circular curve in a horizontal plane. It is provided to avoid sudden change in curvature which may cause discomfort to the passengers and damage to the track.
The formula for the radius of curvature of a transition curve is given by:
r = L²/24h
where,
r = radius of curvature
L = length of transition curve
h = difference in degree of curvature between the main curve and the straight portion
Now, we need to find the relation between r and l for an ideal transition curve.
An ideal transition curve is one in which the rate of change of curvature is constant. It means that the curvature changes uniformly along the transition curve.
Let the degree of curvature at the beginning of the transition curve be D1 and at the end of the transition curve be D2. Then, the difference in degree of curvature, h = D2 - D1.
Let the radius of curvature at any point of the transition curve be r and the distance from the beginning of the transition curve to that point be l.
Using the formula for radius of curvature, we get:
r = L²/24h
Since h is constant for an ideal transition curve, we can write:
r = kL²
where k is a constant.
Differentiating with respect to l, we get:
dr/dl = 2kL
Since the rate of change of curvature is constant for an ideal transition curve, we can write:
dr/dl = constant = kL/2
Solving for r, we get:
r = 1/(kL/2)
r = 2/kL
Comparing with the given options, we see that the correct answer is option 'C':
1/(r) = kL/2
1/r = 2/kL
Transition curve is a curve provided to connect a straight line and a circular curve in a horizontal plane. It is provided to avoid sudden change in curvature which may cause discomfort to the passengers and damage to the track.
The formula for the radius of curvature of a transition curve is given by:
r = L²/24h
where,
r = radius of curvature
L = length of transition curve
h = difference in degree of curvature between the main curve and the straight portion
Now, we need to find the relation between r and l for an ideal transition curve.
An ideal transition curve is one in which the rate of change of curvature is constant. It means that the curvature changes uniformly along the transition curve.
Let the degree of curvature at the beginning of the transition curve be D1 and at the end of the transition curve be D2. Then, the difference in degree of curvature, h = D2 - D1.
Let the radius of curvature at any point of the transition curve be r and the distance from the beginning of the transition curve to that point be l.
Using the formula for radius of curvature, we get:
r = L²/24h
Since h is constant for an ideal transition curve, we can write:
r = kL²
where k is a constant.
Differentiating with respect to l, we get:
dr/dl = 2kL
Since the rate of change of curvature is constant for an ideal transition curve, we can write:
dr/dl = constant = kL/2
Solving for r, we get:
r = 1/(kL/2)
r = 2/kL
Comparing with the given options, we see that the correct answer is option 'C':
1/(r) = kL/2
1/r = 2/kL
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