Q: Suppose a particle moves in a circle with position vector $$\mathbf{r}(t)$$ and angular velocity vector $$\mathbf{\Omega}. $$What is the correct expression for the velocity vector $$\mathbf{v}(t)$$?

$$\mathbf{v}(t) = \mathbf{\Omega} \times \mathbf{r}(t)$$

Question 1

Given the position vector $$\mathbf{r}(t) = t\mathbf{i} + t^2$\mathbf{j} + t^3$\mathbf{k}$$, what is the acceleration vector $$\mathbf{a}(t) at $$any time $$t$$?

Accepted Answer

$$\mathbf{a}(t) = 2\mathbf{j} + 6t\mathbf{k}$$

Question 2

Suppose a particle moves in a circle with position vector $$\mathbf{r}(t)$$ and angular velocity vector $$\mathbf{\Omega}. $$What is the correct expression for the velocity vector $$\mathbf{v}(t)$$?

Accepted Answer

$$\mathbf{v}(t) = \mathbf{\Omega} 	imes \mathbf{r}(t)$$

Question 3

The length of a curve $$\mathbf{r}(t)$$ for $$a \leq t \leq b is $$given by which of the following integrals?

Accepted Answer

$$\int_a^b \left| \frac{d\mathbf{r}}{dt} \right| dt$$

Question 4

If $$\mathbf{u}(t) is a $$differentiable vector function, what is the derivative of its magnitude $$|\mathbf{u}(t)|$$ with respect to $$t$$?

Accepted Answer

$$\frac{d}{dt}|\mathbf{u}(t)| = \frac{\mathbf{u}(t) \cdot \mathbf{u}'(t)}{|\mathbf{u}(t)|}$$

Question 5

A particle moves such that its velocity vector $$\mathbf{v}(t) is $$always perpendicular to its acceleration vector $$\mathbf{a}(t). $$What can be concluded about the speed $$|\mathbf{v}(t)|$$?

Accepted Answer

The speed is constant.

Question 6

Given $$\mathbf{u}(t) is $$three times differentiable, which of the following is the correct expression for $$\frac{d}{dt}\left[\mathbf{u}(t) \cdot \left(\frac{d\mathbf{u}}{dt} 	imes \frac{d^2$\mathbf{u}}{dt^2$}ight)ight]$$?

Accepted Answer

$$\frac{d\mathbf{u}}{dt} \cdot \left(\frac{d\mathbf{u}}{dt} 	imes \frac{d^2$\mathbf{u}}{dt^2$}ight) + \mathbf{u}(t) \cdot \left(\frac{d^2$\mathbf{u}}{dt^2$} 	imes \frac{d^2$\mathbf{u}}{dt^2$}ight) + \mathbf{u}(t) \cdot \left(\frac{d\mathbf{u}}{dt} 	imes \frac{d^3$\mathbf{u}}{dt^3$}ight)$$

Question 7

Which of the following best describes the unit tangent vector $$\hat{T} to a $$curve $$\mathbf{r}(t)$$?

Accepted Answer

$$\hat{T} = \frac{\frac{d\mathbf{r}}{dt}}{\left|\frac{d\mathbf{r}}{dt}\right|}$$

Question 8

The curvature $$\kappa(s) of a $$curve parameterized by arc length $$s is $$defined as:

Accepted Answer

$$\kappa(s) = \left|\frac{d\hat{T}}{ds}\right|$$

Question 9

For a curve in 3D, the unit normal vector $$\hat{N} is $$given by:

Accepted Answer

$$\hat{N} = \frac{d\hat{T}/ds}{\left|d\hat{T}/ds\right|}$$

Question 10

The Frenet-Serret frame at a point on a space curve consists of which three mutually orthogonal unit vectors?

Accepted Answer

The tangent, normal, and binormal vectors

Question 11

If a curve is parameterized by arc length $$s$$, what is the magnitude of the velocity vector $$\mathbf{v}(s)$$?

Accepted Answer

$$|\mathbf{v}(s)| = 1$$

Question 12

The torsion $$	au(s) of a $$space curve measures:

Accepted Answer

The rate at which the curve departs from being planar

Question 13

Which of the following is the correct formula for the binormal vector $$\hat{B} in $$the Frenet frame?

Accepted Answer

$$\hat{B} = \hat{T} 	imes \hat{N}$$

Question 14

A particle moves in a rotating reference frame with angular velocity $$\mathbf{\Omega}. $$Which term represents the Coriolis force per unit mass?

Accepted Answer

$$-2\mathbf{\Omega} 	imes \mathbf{v}$$

Question 15

Given the curve $$\mathbf{r}(t) = t^3$\mathbf{i} + t^2$\mathbf{j}$$ for $$-1 \leq t \leq 2$$, what is the correct integral to compute its total length?

Accepted Answer

$$\int_{-1}^{2} \sqrt{9t^4$ + 4t^2$} dt$$

Question 16

If $$\mathbf{F} = m\mathbf{a} in an $$inertial frame, what is the analogous equation in a rotating frame with angular velocity $$\mathbf{\Omega}$$?

Accepted Answer

$$\mathbf{F} = m[\mathbf{a} + 2\mathbf{\Omega} 	imes \mathbf{v} + \mathbf{\Omega} 	imes (\mathbf{\Omega} 	imes \mathbf{R})]$$

Vector Functions and Space Curves: Calculus II Study Notes

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