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Chapter 13: Problem 2

In each of the following situations, is the work being done positive, negative, or zero? (a) a bull paws the ground; (b) a fishing boat pulls a net through the water behind it; (c) the water resists the motion of the net through it; (d) you stand behind a pickup truck and lower a bale of hay from the truck's bed to the ground. Explain. [Based on a problem by Serway and Faughn.]

Short Answer

Expert verified
(a) Zero work; (b) Positive work; (c) Negative work; (d) Negative work.

Step by step solution

01

Analyzing the Bull Paws

When the bull paws the ground, it is applying a force to the ground. However, if the ground doesn't move in the direction of the force applied by the bull, then no work is done by the bull on the ground according to the work equation \( W = Fd\cos(\theta) \) where \( d \) (distance moved in the direction of force) is zero.
02

Investigating the Fishing Boat

As the fishing boat pulls the net through the water, it exerts a force on the net in the direction of its motion. Work is done since there is a displacement in the direction of the force, resulting in positive work: \( W = Fd \cos(0) = Fd \).
03

Considering Water Resistance

The water exerts a force resisting the motion of the net, which means the force is opposite to the displacement. Thus, the angle \( \theta \) is 180 degrees, giving us \( \cos(180^{\circ}) = -1 \). This results in negative work: \( W = Fd \cos(180^{\circ}) = -Fd \).
04

Lowering the Bale of Hay

When you lower the bale of hay, the force you apply is upwards to counteract gravity, but the displacement of the bale is downwards, resulting in a force opposite to displacement. Therefore, the work done by you on the bale is negative: \( W = Fd \cos(180^{\circ}) = -Fd \).

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Key Concepts

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

Physics of Forces
In physics, a force is an interaction that changes the motion or shape of an object. Forces can cause an object to start moving, stop moving, or change direction. There are different types of forces such as gravitational, electromagnetic, and frictional forces. When discussing work, force is an essential component.

The concept of work involves applying a force to move an object over a distance. The formula for calculating work is \[ W = Fd \cos(\theta) \]where \( W \) is work, \( F \) is the force applied, \( d \) is the displacement of the object, and \( \theta \) is the angle between the force and the displacement.
  • When \( \theta \) is 0 degrees, the force is in the direction of the displacement, resulting in positive work.
  • When \( \theta \) is 180 degrees, the force is opposite to the displacement, resulting in negative work.
  • If there is no displacement, no work is done regardless of the force applied.
For example, when a bull paws at the ground but the ground doesn't move, no work is done despite the force applied.
Displacement in Physics
Displacement refers to the change in position of an object. It is crucial to understand that displacement is different from distance, as it not only considers the length of the path traveled but also the direction. Displacement is a vector quantity: it has both magnitude and direction.

In terms of work, displacement plays a vital role because work is only done when there's movement in the direction of the applied force. For instance, if a force is applied but the object doesn鈥檛 move, the displacement is zero and thus, the work done is also zero.
  • If a fishing boat pulls a net through water and the net moves forward, there is displacement in the direction of the force applied by the fishing boat. Hence, positive work is done on the net.
  • Alternatively, if a force is exerted but the object moves in the opposite direction, the work done is negative, like when water resists the moving net.
Mechanics Problems
Mechanics problems are commonly presented in physics to illustrate the concept of work and energy, highlighting the relationship between forces and motion. Analyzing these problems involves understanding the interaction between different forces and how they influence the movement of objects.

Consider a typical problem where you lower a bale of hay from a truck. As you lower the bale, gravity pulls it downward while you apply an upward force to control the descent. The displacement is downwards, opposite to the force you applied. This means you are performing negative work on the bale.
  • It鈥檚 essential to recognize which forces are doing work in a given situation: the force exerted by an object and the resistance or assistance from external forces.
  • Understanding the angle of the force in relation to the displacement helps determine if the work done is positive, negative, or zero, facilitating a deeper understanding of mechanics problems.
These exercises help students comprehend how applied forces and resulting movements interrelate, thereby better understanding the broader concepts of energy transfer and mechanics.

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Most popular questions from this chapter

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 mass moving in one dimension is attached to a horizontal spring. It slides on the surface below it, with equal coefficients of static and kinetic friction, \(\mu_{k}=\mu_{s} .\) The equilibrium position is \(x=0 .\) If the mass is pulled to some initial position and released from rest, it will complete some number of oscillations before friction brings it to a stop. When released from \(x=a(a>0)\), it completes exactly \(1 / 4\) of an oscillation, i.e., it stops precisely at \(x=0 .\) Similarly, define \(b>0\) as the greatest \(x\) from which it could be released and comlete \(1 / 2\) of an oscillation, stopping on the far side and not coming back toward equilibrium. Find \(b / a\). Hint: To keep the algebra simple, set every fixed parameter of the system equal to \(1 .\)

An airplane flies in the positive direction along the \(x\) axis, through crosswinds that exert a force \(\mathbf{F}=(a+b x) \hat{\mathbf{x}}+(c+d x) \hat{\mathbf{y}}\). Find the work done by the wind on the plane, and by the plane on the wind, in traveling from the origin to position \(x \)

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: $$ \rho(r)=\frac{1}{4 \pi G m r^{2}} \frac{\mathrm{d}}{\mathrm{d} r}\left(r^{2} \frac{\mathrm{d} P E}{\mathrm{~d} r}\right) $$

Two speedboats are identical, but one has more people aboard than the other. Although the total masses of the two boats are unequal, suppose that they happen to have the same kinetic energy. In a boat, as in a car, it's important to be able to stop in time to avoid hitting things. (a) If the frictional force from the water is the same in both cases, how will the boats' stopping distances compare? Explain. (b) Compare the times required for the boats to stop.
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