Chapter 4: Work and energy (C8502200)

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1 Work

Work (done) occurs when a force causes displacement of a body, defined as $W=Fcdot d=F.d.cos( heta)$. In contrast with torque which required the consideration of displacement perpendicular to its force, work only considers displacement parallel to its force, and hence requires the use of the $cos$ function. Its SI units is joules, and it is a scalar quantity.

Little did Jamie or Mandy know that they were both working for, ‘The Agency’. Coincidently, they were both working on computers in the SIGINT wing of The Agency, not realizing they’d been right next to another, for hours.

“No such thing as coincidence!” Mandy remarked.

“Oh, hi,” they both remarked as they looked at another.

“Oh my goodness,” they both continued, in unison.

“I still remember you had Open Your Eyes to Love playing in the background, just to make it more awkward .”

“Jamie, right?” Mandy remarked.

“Yeah,” Jamie replied, “Mandy, right?”

“Yeah, I can’t believe you work here,” Jamie replied.

“Hey,” Mandy said, trying to find a way to lure him, “I’m going to this thing tonight, do you want to come?”

“Umm,” Jamie thought, “what is it?”

“Umm,” Mandy replied, unsure whether this was a good idea, “we’re doing a bible study at Mr. Whittaker’s.”

“Oh, I’m Christian too,” Jamie replied, “sure.”

As Jamie left the room, he tossed his unfinished energy drink into the bin. Mandy, filled dazed to the brink, took the drink back out, thought about it, and drank it.

“That was very classy,” Jamie remarked.

“We were nineteen,” Mandy retorted.

 Formative learning activity Maps to RK4.A What is work?

2 Energy

Energy, though not definable, is the ability to do work. This is not entirely correct however, because some systems have energy, but still can’t do work. Alternatively, because work done requires energy, it can be said that work is the transfer of energy. Its SI units is joules (thereby being the same unit as work), or eV. An eV is $1.6 imes 10^-19 J$, hence only used on a subatomic level. Energy is a scalar quantity, hence should be preferred as a calculation method than using vectors. Mechanical energy can relate to either:

• Kinetic energy, which is energy due to motion, $KE=dfrac{1}{2}mv^2$
• Potential energy, which is energy due to position of a body:
• Gravitational potential energy, which is energy due to elevated positions, and for positions near the earth surface is $PE=E=Fd=mag$. Note that due $h$ is the distance above an arbitrary point, potential energy defined in this manner as it isn’t an absolute value, but rather, a change in energy. For objects not positioned near earth, $F=dfrac{GMm}{r^2}$ with $W=Fd$ can be used instead. Work has the same units as energy. Work is thus $W=Fd=Fr$ (since $r$ is same as $d$), $W=dfrac{GMm}{r^2}.r=dfrac{GMm}{r}$. In reality, the formula should be $PE=-dfrac{GMm}{r}$, since energy is required to separate objects against gravitational pull, thereby increasing potential energy, when radius is increased (without the negative, as radius is increased, potential energy decreases, as they are inversely proportional). In other words, at infinite distance, objects have a gravitational energy of $0J$, and this number becomes increasingly negative as they come closer to each other
• Elastic potential energy, which is energy due to an object under tension or compression, and is per Hooke’s law, $F=-kDelta x$. Integrating to find work, $PE=dfrac{1}{2}kDelta x^2$
• Electrostatic potential energy, discussed

Thermodynamic system is a macroscopic region, defined by boundaries which separate it from its surroundings. These include:

• Isolated systems, which don’t permit the exchange of mass or energy with its surroundings. For example, the entire universe is isolated, as no mass nor energy can enter nor escape
• Closed system, which permit energy (but not mass) to be exchanged with its surroundings. For example, a tennis ball is a closed system, as energy is transferred into the ball when it is dropped, but the mass of the ball remains constant
• Open system, which permit both energy and mass to be exchanged with its surroundings. For example, a pipe with fluid flowing through, transfers mass and energy of fluid in and out of a pipe

The 1st law of thermodynamics, is the law of conservation of energy in an isolated system, $Delta E=0$. In a closed system, as energy can now be exchanged, by either supplying energy or heat into the system, $Delta E=W+q$, where $W$ is the amount of work done on the system, and $q$ is the amount of heat supplied into the system. As defined , work is the transfer of energy via a force, thereby resulting in movement. Heat is the transfer of energy from a hotter to colder body. For example, rubbing materials together thereby inducing heat, is not considered heat, but rather, work, as there is no hotter or colder body.

Physics is generally unconcerned with heat (heat will be iterated on ), so assuming there is no heat, the 1stlaw of thermodynamics can be rewritten as $Delta E=W$. As the energies present include the kinetic and potential energies, $Delta E=Delta KE+Delta PE=W$. Where there is also no change in potential energy (i.e. neither getting closer to nor further away from earth, for example in the case of a hockey puck), $W=Delta KE=K_{t}-K_{0}$, known as the work-energy theorem. The 1stlaw of thermodynamics explains why the formula $W=Fcdot d$ doesn’t apply for friction, since for friction, $E=F.d$, and given the 1st law of thermodynamics is $Delta E=W+q$, this means $W+q=F.d$, or reshuffling, $W=F.d-q$.

Conservative forces are forces, which have a work done between two points, independent of the path taken. If a conservative force is applied to a particle which travels away and back to its point of origin, the work done should be thus $0$. Conservative forces can be identified as having a potential energy associated. For example, gravitational forces which have an associated gravitational potential energy, meaning gravity is a conservative force. In contrast, friction is non-conservative, as it has no potential energy associated with it. Also, conservative forces are dependent on distance, for example,  makes consideration for distance $F=dfrac{GMm}{r^2}$. Conservative forces are conservative, because it conserves mechanical energy. If total energy doesn’t change, there isn’t ability to do work (definition of energy). Where there is implication work done is done, for example, gravity moving an object towards earth, what energy is implied as meaning, is actually change in potential or kinetic energy, as supposed to change in total energy (which there is none).

Power is the rate at which energy is used [or transferred], and uses the SI units watt, or joules per second. It is classically defined as $P=dfrac{W}{t}$. Note however, that by the 1stlaw of thermodynamics,  $Delta E=W+q$. The formula should thus be $P=dfrac{E}{t}=dfrac{W+q}{t}$, meaning the classical definition only holds if there is no heat. Power in a mechanical system can also be defined as $P=Fcdot v=F.v.cos( heta)$ (remembering the dot product or $cos( heta)$ achieves multiplication of the components of the vectors parallel to each other).

 Formative learning activity Maps to RK4.B What is energy?

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