Task 5.1

Work is a transfer of energy. When a force acts through a displacement, work is done. Gravity acts on falling objects, transferring potential energy to kinetic energy. Any force, when acting through a displacement, will result work done.

  • When a constant force and displacement are in the same straight line direction, the work done is simply the product of the magnitude of the force, and the distance.
  • When a constant force acts opposite a straight line displacement, the work is the negative of the magnitude of the force and the distance.

If a constant force is not parallel (or antiparallel) to a straight line displacement $\vec d$, then we instead use the component of the force that is parallel to the displacement (so $\vec F_{\parallel \vec d}$) to compute work.

Let $\vec F=(-1,2)$ and $\vec d=(3,4)$.

  1. Start by computing $\vec F_{\parallel d} = \text{proj}_{\vec d}\vec F$ and $\vec F_{\perp d}$.
  2. Construct a picture that shows the relationship between $\vec F,$ $\vec d$, $\text{proj}_{\vec d}\vec F$, and $\vec F_{\perp d}$.
  3. Compute the work done by $\vec F$ through the displacement $\vec d$ by computing $|\vec F_{\parallel d}|$ and $|\vec d|$. Should the work be positive or negative?

Change the force to $\vec F = (-2,0)$. but keep $\vec d=(3,4)$.

  1. Construct a similar picture as above, showing the relationship between $\vec F,$ $\vec d$, $\text{proj}_{\vec d}\vec F$, and $\vec F_{\perp d}$. Feel free to construct this picture with, or without, doing any computations.
  2. Compute the work done by $\vec F$ through the displacment $\vec d$. Should the work be positive or negative?
  3. Can you find a simpler way to compute the work done by $\vec F$ through $\vec d$ than computing $|\vec F_{\parallel d}|$ and $|\vec d|$?

Task 5.2

  1. Find the length of the curve $\ds \vec r(t) = \left(t^3,\frac{3t^2}{2}\right)$ for $t\in[1,3]$. The notation $t\in[1,3]$ means $1\leq t\leq 3$. Be prepared to show us your integration steps in class (you'll need a substitution).
  2. Now find the length of the helix $\vec r(t) = (2\cos t, 2\sin t, t)$ for $t\in [0, 4\pi] $.

Task 5.3

Suppose a rover is currently moving and has a velocity vector $\vec v = (3,4)$. A force acts on the rover causing an acceleration of $\vec a = (-1,5)$. The rover is currently at the location $(2,-3)$.

  1. Draw picture that shows the rover's location along with the velocity and acceleration vectors drawn with their base at the rover's location.
  2. Find the vector component of the acceleration that is parallel to the velocity (so find $\vec a_{\parallel \vec v}$), and then find the vector component of the acceleration that is orthogonal to the velocity (so find $\vec a_{\perp \vec v}$).
  3. Will this acceleration cause the rover to speed up or slow down? Explain.
  4. Will this acceleration cause the rover to turn left or right? Explain.

A probe above Mars is currently moving and has a velocity vector $\vec v = (-2,1,2)$. The onboard thrusters apply a force that causes an acceleration of $\vec a = (0,2,-3)$.

  1. Find both $\vec a_{\parallel \vec v}$ and $\vec a_{\perp \vec v}$.
  2. Will this acceleration cause the satellite to speed up or slow down? Explain.
  3. How would you interpret $\vec a_{\perp \vec v}$?

Task 5.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.

  • Work: section 2.3, checkpoint 2.29 and exercises 175-179
  • Projection: section 2.3, checkpoint 2.27 and exercises 167-172
  • Arc Length: section 3.3: checkpoint 3.9, exercises 102-112