Q: For a curve $$\mathbf{r}(t)$$, the curvature $$\kappa$$ can be computed using which of the following formulas?

$$\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3}$$

Q: Which of the following is the correct formula for the torsion $$\tau of a $$curve $$\mathbf{r}(t)$$?

$$\tau = \frac{\det \begin{pmatrix} \dot{x} & \dot{y} & \dot{z} \\ \ddot{x} & \ddot{y} & \ddot{z} \\ \dddot{x} & \dddot{y} & \dddot{z} \end{pmatrix}}{|\mathbf{v} \times \mathbf{a}|^2}$$

Question 1

Which of the following best describes the binormal vector $$\mathbf{B} at a $$point on a space curve?

Accepted Answer

$$\mathbf{B} is $$the cross product of the tangent and normal vectors: $$\mathbf{B} = \mathbf{T} 	imes \mathbf{N}.$$

Question 2

Given a particle moving along a curve with position vector $$\mathbf{r}(t)$$, which formula gives the tangential component of acceleration?

Accepted Answer

$$a_T = \frac{d}{dt}|\mathbf{v}|$$

Question 3

If a car moves around a circular track of radius $$R at $$constant speed $$v$$, what is the normal component of its acceleration?

Accepted Answer

$$a_N = \frac{v^2$}{R}$$

Question 4

Which of the following statements about the TNB (Frenet) frame is correct?

Accepted Answer

The vectors $$\mathbf{T}$$, $$\mathbf{N}$$, and $$\mathbf{B}$$ form a right-handed orthonormal frame moving along the curve.

Question 5

For a curve $$\mathbf{r}(t)$$, the curvature $$\kappa$$ can be computed using which of the following formulas?

Accepted Answer

$$\kappa = \frac{|\mathbf{v} 	imes \mathbf{a}|}{|\mathbf{v}|^3}$$

Question 6

A particle moves along a curve with velocity $$\mathbf{v}$$ and acceleration $$\mathbf{a}. If$$ $$a_T = 0$$, what does this imply about the motion?

Accepted Answer

The speed of the particle is constant.

Question 7

Which of the following is the correct formula for the torsion $$	au of a $$curve $$\mathbf{r}(t)$$?

Accepted Answer

$$	au = \frac{\det \begin{pmatrix} \dot{x} & \dot{y} & \dot{z} \ \ddot{x} & \ddot{y} & \ddot{z} \ \dddot{x} & \dddot{y} & \dddot{z} \end{pmatrix}}{|\mathbf{v} 	imes \mathbf{a}|^2}$$

Question 8

Suppose a roller coaster moves along a track with high curvature and high speed at a certain point. What can you conclude about the normal component of acceleration at that point?

Accepted Answer

It is large, since $$a_N = \kappa |\mathbf{v}|^2.$$

Question 9

If the acceleration vector $$\mathbf{a} of a $$particle is always orthogonal to its velocity $$\mathbf{v}$$, what can you say about the motion?

Accepted Answer

The speed of the particle is constant.

Question 10

A vertex on a curve in space is defined as a point where:

Accepted Answer

The torsion $$	au is $$zero.

Question 11

Given $$\mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N}$$, which formula allows you to compute $$a_N$$ without knowing the curvature?

Accepted Answer

$$a_N = \sqrt{|\mathbf{a}|^2 - a_T^2$}$$

Question 12

Which of the following best describes the osculating plane at a point on a curve?

Accepted Answer

It is the plane determined by the tangent and normal vectors.

Question 13

A particle moves along a helix $$\mathbf{r}(t) = a \cos t, a \sin t, bt. $$What is the curvature $$\kappa$$?

Accepted Answer

$$\kappa = \frac{a}{a^2$ + b^2$}$$

Question 14

Which of the following is true about the acceleration vector $$\mathbf{a}$$ for a particle moving in space?

Accepted Answer

$$\mathbf{a}$$ always lies in the plane determined by $$\mathbf{T}$$ and $$\mathbf{N}.$$

Question 15

If a particle's speed doubles while moving along a curve of fixed curvature $$\kappa$$, what happens to the normal component of acceleration?

Accepted Answer

It increases by a factor of 4.

Question 16

Which of the following is NOT a property of the binormal vector $$\mathbf{B}$$?

Accepted Answer

$$\mathbf{B} is $$always parallel to the tangent vector.

Question 17

The normal plane at a point on a curve is defined as:

Accepted Answer

The plane determined by the normal and binormal vectors.

Question 18

Which of the following real-world scenarios best illustrates the importance of the normal component of acceleration?

Accepted Answer

A car making a sharp turn at high speed, where passengers feel pushed outward.

Tangential and Normal Components of Acceleration, Curvature, and Torsion in Space Curves

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