In this posting, we study the concept of work. You may find more detailed exposition with various examples in Section 8.5 of your book.
Slogan: Work is "energy".
Slogan: Work is "energy".
Linear fashion (local/micro work). Work = Force $\times$ Distance
Nonlinear fashion (global/total work). 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. This means that any object has a constant acceleration, not depending on the object, 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 estimated 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. However, this will rarely happen. This formula will usually make sense when we compute micro-force.
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. However, this will rarely happen. This formula will usually make sense when we compute micro-force.
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 meters.)
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 meters.)
(Hint: This is a very hard exercise for any student who is taking Math 116 from my past experience. Writing $g$ for the constant gravitational acceleration, 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. Note that by taking derivative, you see $\tan(\theta(x)) = 2x,$ and this would let you compute the integral.)
Remark. The unit for the work in the last exercise is in joules (i.e., $kg \cdot m^{2}/s^{2}$, which is often denoted as $J$).
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.
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 meters.)
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 meters.)
(Hint: This is a very hard exercise for any student who is taking Math 116 from my past experience. Writing $g$ for the constant gravitational acceleration, 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. Note that by taking derivative, you see $\tan(\theta(x)) = 2x,$ and this would let you compute the integral.)
Remark. The unit for the work in the last exercise is in joules (i.e., $kg \cdot m^{2}/s^{2}$, which is often denoted as $J$).
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.
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