A measure of how quickly a curve bends or deviates from being a straight line at a given point.
A tight turn has high curvature; a gentle bend has low curvature.
Showing a random 20 of 50 problems.
Example 1
easy
True or false: doubling a circle's radius halves its curvature.
Example 2
medium
A road designer wants a constant centripetal-feel curve. Which shape should the road follow and why?
Example 3
medium
A satellite orbits Earth at altitude where the orbit radius is 7000 km. What is the curvature of the orbit?
Example 4
medium
A turning radius restriction posts ρ≥50 m. What is the largest allowed curvature?
Example 5
medium
At a point on a curve, the 'best-fitting circle' has radius 5. What is the curve's curvature there?Osculating (best-fitting) circle has radius r = 5 — find κ = 1/r
Example 6
easy
Two circles have radii 10 cm and 40 cm. What is the ratio of their curvatures (small to large)?
Example 7
challenge
A sphere of radius R — how does its surface curvature compare to a circle of radius R, qualitatively, and why does this matter for flat maps of Earth?
Example 8
easy
Which has greater curvature: a circle of radius 7 or a circle of radius 0.5?
Example 9
medium
Order from least to greatest curvature: a straight road, a gentle highway curve (r = 500 m), a roundabout (r = 20 m).
Example 10
hard
At a point on an ellipse, the osculating circle has radius r1 at the end of the major axis and r2 at the end of the minor axis. For a 4×2 ellipse (semi-major 4, semi-minor 2), r1=ab2=1 and r2=ba2=8. Which end has higher curvature?
Example 11
medium
Why does a larger circle have SMALLER curvature, even though it's bigger?
Example 12
hard
At a sharp bend, the osculating circle has radius 0.25 m. The car's speed is 5 m/s. Find the centripetal acceleration using a=κv2.
Example 13
hard
A truck's safe maximum centripetal acceleration is 3 m/s2. At speed 15 m/s, what is the maximum curvature the road can have?
Example 14
challenge
Why does a highway use a gradual 'spiral' transition between a straight section and a circular curve, in terms of curvature?
Example 15
medium
Why does an ellipse NOT have constant curvature, while a circle does?
Example 16
easy
A bicycle wheel has radius 0.3 m. What is the curvature of the rim?
Example 17
easy
A circle has curvature 61. What is its radius?Given κ = 1/6, find radius r = 1/κ
Example 18
easy
A circle has radius 4. What is its curvature?Curvature κ = 1/r = 1/4 for a circle of radius 4
Example 19
easy
A circle has radius r=4 cm. What is its curvature κ? Compare with a circle of radius r=1 cm.Circle of radius r = 4 has curvature κ = 1/r = 1/4