Class 9 - Work (9/22/2017, Friday)

Deadline to drop

Sept 25, Monday: Keep in mind that Math 116 will require you a significant amount of work outside the class! It won't be possible, even if you had good mathematical background, to just sit in the courses without work.

1st Gateway

The first Gateway will close on Sept 25, Monday. Please take it because otherwise there is a huge penalty on your final letter grade.

2nd Gateway

The second Gateway will start on Sept 26, Tuesday and you will earn extra 10% quiz points if you pass it by Sept 29, Friday. (Of course, 20% total if you passed 1st Gateway early.)

Webwork

• 7.5 due 9/25 Mon

Now mathematics. We will roughly cover Section 8.5 of your book.

Work: physical quantity representing "energy"

Linear fashion. Work = Force x Distance

Nonlinear fashion. When $x$ is a position that moves from $a$ to $b$ and $F(x)$ is the force acting at $x$, then

$$\text{Work} = \int_{a}^{b}F(x) dx.$$

Gravitational acceleration from physics. It is quite a remarkable observation that on Earth, if you drop any two objects from the same height when there is no air at presence, both objects will hit the ground at the same time. One more remarkable observation is that any object has a constant acceleration when we do not count air resistance. In the unit (meter/second)/second = $m/s^{2}$, this constant is denoted as $g$, and it is known that it is approximately equal to $9.8$, so when the problem does not give you any information about gravitation, you should start by write something like "Let $g$ be the gravitational acceleration in $m/s^{2}$" or "Let $g = 9.8 m/s^{2}$ be the gravitational constant."

Force is mass times acceleration when the quantities are constant. In classical mechanics, it is assumed that the force is equal to mass times acceleration (a.k.a., the "law" $F = ma$). This is something we assume in this course as well, if the mass and acceleration are assumed to be constant.

Exercise. Suppose that the mass of your book is $2$kg. If you have lifted your book $1.5$ meters off the floor, then how much work have you done?

Exercise. Say you are pushing a ball whose mass is $1$kg along the graph $y = x$ from point $(0, 0)$ to $(1, 1)$. When you are done, how much work would you have done? (Suppose that the unit for distance on our $xy$-plane is in meter.)

Exercise (Hard). Say you are pushing a ball whose mass is $1$kg along the graph $y = x^{2}$ from point $(0, 0)$ to $(1, 1)$. When you are done, how much work would you have done? (Suppose that the unit for distance on our $xy$-plane is in meter.)

This is a very hard exercise for any student who is taking Math 116 from my past experience. Writing $g m/s^{2}$ as gravitational acceleration, which we assume to be constant, if you compute correctly the instantaneous force for the particle with $x$-coordinate $x$ is equal to $g \sin(\theta(x))$ and the instantaneous distance would be $\Delta x / \cos(\theta(x))$. Here, note that $\theta(x)$ is the angle between the tangent line of the point $(x, y)$ on the graph $y = x^{2}$ and the $x$-axis. The unit for work is in joules (i.e., $kg \cdot m^{2}/s^{2}$, which is often denoted as $J$).

Note that by taking derivative, you see $\tan(\theta(x)) = 2x,$ and this would let you compute the integral.

Exercise*. Solve 3a and 3b of Exam 1, Winter 2017.

Exercise*. Solve 9a, b, and c of Exam 1, Winter 2017.

Exercise*. Solve 9 of Exam 1, Winter 2016.

Exercise*. Solve 9 of Exam 1, Fall 2015.