Arc Length and Curvature

How long the path is, and how sharply it bends.

12 problems · about 23 minutes · N-VM.A.1

NameDate

Figure — use these to answer the problems

Widen the parabola with a and the bend at the bottom softens. That is curvature falling.

y = 1x² + 0x + 0

a, how sharply it bends1

point on the curve0

Figure 1

Straightforward practice. Get the method working first.

5 problems

A circle of radius 25. What is its curvature, as a decimal?
Answer
2
What is being integrated in an arc length integral?
- The speed, which is the magnitude of the velocity.
- The position vector itself.
Speed is constant at 9 from t = 0 to t = 4. What is the arc length?
Answer
A circle of radius 4. What is its curvature, as a decimal?
Answer
A straight line. What is its curvature?
Answer

The same ideas with more to keep track of.

3 problems

Curvature is 0.2. What is the radius of curvature?
Answer
r(t) = ⟨5t, 12t, 0⟩ from t = 0 to t = 3. What is the arc length?
Answer
Which bends more sharply, a circle of radius 2 or one of radius 10? Enter the radius of the sharper one.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Order these paths from least curved to most curved at their tightest point.
Write 1 to 4 in the boxes to put these in order.
- A straight road
- A gentle motorway curve of radius 500 m
- A roundabout of radius 20 m
- A hairpin bend of radius 5 m
The speed of a particle is 0 for a whole interval. What is the arc length over that interval?
Answer
The Track
A cart moves at constant speed 6 for 7 seconds. How far does it travel?
Answer
The Bend
A circular bend of radius 20 metres. What is its curvature, as a decimal?
Answer