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Chapter 13: Problem 18

Does it make sense to say that work is conserved?

Short Answer

Expert verified
No, work is not conserved because it represents energy transfer, not a quantity that remains constant.

Step by step solution

01

Defining Work

Work in physics is defined as the energy transferred to an object via a force causing displacement. It is given by the formula \( W = F imes d imes \cos(\theta) \) where \( W \) is work, \( F \) is the force applied, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors.
02

Understanding Conservation in Physics

The concept of conservation in physics refers to a quantity that remains constant within an isolated system over time. Common conserved quantities include energy, momentum, and mass.
03

Work versus Energy

It is crucial to note that while energy is conserved within a closed system, work is not. Work is a measure of energy transfer, not a quantity that is independently conserved. The energy within a system can change due to external forces doing work on the system or the system doing work on its surroundings.
04

Relationship Between Work and Energy Conservation

When work is done on a system, the energy of the system changes, which reflects the conservation of energy principle, not work. For example, if work is done against friction, energy within a system might convert from kinetic to thermal energy.
05

Conclusion

Work is not a conserved quantity because it represents energy transfer between systems or different forms within a system. Therefore, it changes depending on the interactions occurring, affirming that only the total energy within a system is conserved, not work as an isolated concept.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Work in Physics
In physics, work is an essential concept that describes the transfer of energy via forces. When you apply a force to move an object, you're doing work on that object. This transfer of energy helps us quantify the effort involved in moving things around us. The calculation of work is given by the formula \[ W = F \times d \times \cos(\theta) \]where:
  • \( W \) is the work done,
  • \( F \) is the magnitude of the force applied,
  • \( d \) is the displacement of the object,
  • \( \theta \) is the angle between the force and the displacement vectors.
This equation tells us that work is only done when the force and displacement occur in the same direction, or at least have a component acting in the same direction.
It's also important to note that without displacement, work is zero. This means no matter how much force you exert on an object, if it doesn't move, technically, no work is done.
Energy Transfer
Energy transfer in physics is the movement of energy from one place, object, or form to another. This process is fundamental to our understanding of how energy works and affects objects. There are various forms of energy, including kinetic, potential, and thermal energy.

Whenever work is done, energy is transferred. For example:
  • When you push a book across a table, your hand applies force, transferring energy to the book.
  • The book's energy might change from potential to kinetic as it begins to move.
  • If it slides to a stop due to friction, some energy converts to thermal energy due to heat generated.
This continuous flow and transformation of energy demonstrate what's known as energy conservation, which states energy cannot be created or destroyed but only transformed or transferred. In a mechanical context, understanding energy transfer helps us grasp how systems function and analyze how energy flows and transforms within different contexts.
Closed System
A closed system in physics is one where, ideally, no matter or energy enters or exits the system's boundaries. This isolation allows us to better examine the laws of physics such as the conservation of energy.

Understanding closed systems is crucial for analyzing how energy behaves. Key points about closed systems include:
  • Energy within a closed system remains constant over time if no external work is done.
  • Energy can, however, change forms, exemplified by kinetic energy converting to potential energy.
  • For real-world applications, considering a nearly closed system helps approximate these changes in energy.
Despite being idealized, closed systems provide insight into the conservation laws during an energy transfer. This ensures any transformation or transition of energy types remains consistent with the principle of energy conservation.

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Most popular questions from this chapter

Weiping lifts a rock with a weight of \(1.0 \mathrm{~N}\) through a height of \(1.0 \mathrm{~m}\), and then lowers it back down to the starting point. Bubba pushes a table \(1.0 \mathrm{~m}\) across the floor at constant speed, requiring a force of \(1.0 \mathrm{~N}\), and then pushes it back to where it started. (a) Compare the total work done by Weiping and Bubba. (b) Check that your answers to part a make sense, using the definition of work: work is the transfer of energy. In your answer, you'll need to discuss what specific type of energy is involved in each case.

In the power stroke of a car's gasoline engine, the fuel-air mixture is ignited by the spark plug, explodes, and pushes the piston out. The exploding mixture's force on the piston head is greatest at the beginning of the explosion, and decreases as the mixture expands. It can be approximated by \(F=a / x\), where \(x\) is the distance from the cylinder to the piston head, and \(a\) is a constant with units of \(\mathrm{N} \cdot \mathrm{m}\). (Actually \(a / x^{1.4}\) would be more accurate, but the problem works out more nicely with \(a / x !\) ) The piston begins its stroke at \(x=x_{1}\), and ends at \(x=x_{2}\). The 1965 Rambler had six cylinders, each with \(a=220 \mathrm{~N} \cdot \mathrm{m}, x_{1}=1.2 \mathrm{~cm}\), and \(x_{2}=10.2 \mathrm{~cm}\) (a) Draw a neat, accurate graph of \(F\) vs \(x\), on graph paper. (b) From the area under the curve, derive the amount of work done in one stroke by one cylinder. (c) Assume the engine is running at 4800 r.p.m., so that during one minute, each of the six cylinders performs 2400 power strokes. (Power strokes only happen every other revolution.) Find the engine's power, in units of horsepower \((1 \mathrm{hp}=746 \mathrm{~W})\) (d) The compression ratio of an engine is defined as \(x_{2} / x_{1}\). Explain in words why the car's power would be exactly the same if \(x_{1}\) and \(x_{2}\) were, say, halved or tripled, maintaining the same compression ratio of \(8.5 .\) Explain why this would not quite be true with the more realistic force equation \(F=a / x^{1.4}\).

A certain binary star system consists of two stars with masses \(m_{1}\) and \(m_{2}\), separated by a distance \(b\). A comet, originally nearly at rest in deep space, drops into the system and at a certain point in time arrives at the midpoint between the two stars. For that moment in time, find its velocity, \(v\), symbolically in terms of \(b, m_{1}\), \(m_{2}\), and fundamental constants.

A space probe of mass \(m\) is dropped into a previously unexplored spherical cloud of gas and dust, and accelerates toward the center of the cloud under the influence of the cloud's gravity. Measurements of its velocity allow its potential energy, \(P E\), to be determined as a function of the distance \(r\) from the cloud's center. The mass in the cloud is distributed in a spherically symmetric way, so its density, \(\rho(r)\), depends only on \(r\) and not on the angular coordinates. Show that by finding \(P E\), one can infer \(\rho(r)\) as follows: $$ \rho(r)=\frac{1}{4 \pi G m r^{2}} \frac{\mathrm{d}}{\mathrm{d} r}\left(r^{2} \frac{\mathrm{d} P E}{\mathrm{~d} r}\right) $$

"Big wall" climbing is a specialized type of rock climbing that involves going up tall cliffs such as the ones in Yosemite, usually with the climbers spending at least one night sleeping on a natural ledge or an artificial "portaledge." In this style of climbing, each pitch of the climb involves strenuously hauling up several heavy bags of gear a fact that has caused these climbs to be referred to as "vertical ditch digging." (a) If an \(80 \mathrm{~kg}\) haul bag has to be pulled up the full length of a \(60 \mathrm{~m}\) rope, how much work is done? (b) Since it can be difficult to lift \(80 \mathrm{~kg}\), a \(2: 1\) pulley is often used. The hauler then lifts the equivalent of \(40 \mathrm{~kg}\), but has to pull in \(120 \mathrm{~m}\) of rope. How much work is done in this case?
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