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Chapter 8: Problem 26

At time \(t_{i},\) the kinetic energy of a particle is \(30.0 \mathrm{J}\) and the potential energy of the system to which it belongs is \(10.0 \mathrm{J}\) At some later time \(t\), the kinetic energy of the particle is 18.0 J. (a) If only conservative forces act on the particle, what are the potential energy and the total energy at time \(t_{f} ?\) (b) If the potential energy of the system at time \(t_{f}\) is \(5.00 \mathrm{J},\) are there any nonconservative forces acting on the particle? Explain.

Short Answer

Expert verified
In case (a), the potential energy at \(t_{f}\) is 22.0 J and the total energy is 40.0 J. In case (b), the system has nonconservative forces acting on it as the total energy (23.0 J) at \(t_{f}\) is less than the initial total energy.

Step by step solution

01

Calculation of initial total energy

The total energy of the particle at time \(t_{i}\) is given as the sum of kinetic and potential energy. This can be computed as \( E_{i} = K_{i} + U_{i} = 30.0 \, \mathrm{J} + 10.0 \, \mathrm{J} = 40.0 \, \mathrm{J}\).
02

Calculation of final total energy and potential energy under conservative forces

If only conservative forces are acting, the total energy should be conserved. Therefore, the total energy at time \(t_{f}\) is equal to the intrinsic total energy, i.e., \(E_{f} = E_{i} = 40.0 \, \mathrm{J}\). The kinetic energy at time \(t_{f}\) is given as 18.0 J, therefore the potential energy at time \(t_{f}\) can be calculated by subtracting the kinetic energy from the total energy: \(U_{f} = E_{f} - K_{f} = 40.0 \, \mathrm{J} - 18.0 \, \mathrm{J} = 22.0 \, \mathrm{J}\).
03

Determination of presence of nonconservative forces

In the second scenario, the potential energy at time \(t_{f}\) is given as 5.0 J. Let's calculate the total energy: \(E'_{f} = K_{f} + U'_{f} = 18.0 \, \mathrm{J} + 5.0 \, \mathrm{J} = 23.0 \, \mathrm{J}\). We can see that \(E'_{f}\) is less than \(E_{i}\), so the total energy is not conserved. Thus, nonconservative forces must be acting on the particle because only then would the total energy of the system decrease.

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Key Concepts

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

Understanding Kinetic Energy
Kinetic energy is a fundamental concept in physics that deals with the energy possessed by an object due to its motion. It's usually calculated using the equation \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the particle. In this exercise, we are told that the kinetic energy changes from 30.0 J to 18.0 J, indicating a change in the motion of the particle. Here are some key points to remember about kinetic energy when solving these types of problems:
  • Kinetic energy is always a positive quantity because it depends on the square of the velocity.
  • A decrease in kinetic energy typically means the object is slowing down unless it has negative work done on it.
  • In the context of energy conservation, a change in kinetic energy is often accompanied by a change in potential energy if no nonconservative forces are acting.
By looking at how kinetic energy changes, we can infer the interactions of forces at play.
Exploring Potential Energy
Potential energy represents the stored energy of a system due to its position or configuration. In this exercise, it plays a crucial role in understanding the energy dynamics.The potential energy in a system like this is often calculated with respect to conservative forces, like gravity or spring forces. At time \( t_i \), the potential energy is 10.0 J, while at time \( t_f \), under conservative forces, it's 22.0 J. When we consider potential energy:
  • It is associated with forces that have the potential to do work, such as gravitational force or elastic force in springs.
  • A change in potential energy can lead to a change in kinetic energy, maintaining total energy conservation in a system with only conservative forces.
  • This exercise shows how potential energy can increase when kinetic energy decreases, sustaining the total energy balance if only conservative forces are involved.
Potential energy is crucial for calculating the total mechanical energy and analyzing systems in static equilibrium.
Conservative Forces in Mechanics
Conservative forces are forces where the work done is independent of the path taken by the particle, being a function of initial and final positions alone. Examples include gravitational and elastic forces. In this problem, understanding conservative forces is key to analyzing energy dynamics.In the context of this exercise, assuming conservative forces ensures that total mechanical energy (sum of kinetic and potential energy) is constant. Here are important considerations:
  • With only conservative forces acting, the sum of kinetic and potential energy remains unchanged over time.
  • If the observed total energy changes (as seen when the potential energy at \( t_f \) is given as 5.0 J), nonconservative forces, like friction or air resistance, must be acting on the system.
  • Nonconservative forces cause energy dissipation, usually converting mechanical energy into forms like thermal energy, thus reducing the total mechanical energy.
Recognizing the type of forces involved helps predict changes in the system's energy and explain energy dissipation scenarios effectively.

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Most popular questions from this chapter

The mass of a car is 1500 kg. The shape of the body is such that its aerodynamic drag coefficient is \(D=0.330\) and the frontal area is \(2.50 \mathrm{m}^{2} .\) Assuming that the drag force is proportional to \(v^{2}\) and neglecting other sources of friction, calculate the power required to maintain a speed of \(100 \mathrm{km} / \mathrm{h}\) as the car climbs a long hill sloping at \(3.20^{\circ}\).

A \(70.0-\mathrm{kg}\) diver steps off a \(10.0-\mathrm{m}\) tower and drops straight down into the water. If he comes to rest \(5.00 \mathrm{m}\) beneath the surface of the water, determine the average resistance force exerted by the water on the diver.

A uniform chain of length \(8.00 \mathrm{m}\) initially lies stretched out on a horizontal table. (a) If the coefficient of static friction between chain and table is \(0.600,\) show that the chain will begin to slide off the table if at least 3.00 m of it hangs over the edge of the table. (b) Determine the speed of the chain as all of it leaves the table, given that the coefficient of kinetic friction between the chain and the table is \(0.400 .\)

A \(75.0-\mathrm{kg}\) skysurfer is falling straight down with terminal speed \(60.0 \mathrm{m} / \mathrm{s} .\) Determine the rate at which the skysurfer- Earth system is losing mechanical energy.

At 11: 00 A.M. on September \(7,2001,\) more than 1 million British school children jumped up and down for one minute. The curriculum focus of the "Giant Jump" was on earthquakes, but it was integrated with many other topics, such as exercise, geography, cooperation, testing hypotheses, and setting world records. Children built their own seismographs, which registered local effects. (a) Find the mechanical energy released in the experiment. Assume that 1050 000 children of average mass \(36.0 \mathrm{kg}\) jump twelve times each, raising their centers of mass by \(25.0 \mathrm{cm}\) each time and briefly resting between one jump and the next. The free-fall acceleration in Britain is \(9.81 \mathrm{m} / \mathrm{s}^{2},\) (b) Most of the energy is converted very rapidly into internal energy within the bodies of the children and the floors of the school buildings. Of the energy that propagates into the ground, most produces high-frequency "microtremor" vibrations that are rapidly damped and cannot travel far. Assume that \(0.01 \%\) of the energy is carried away by a long-range seismic wave. The magnitude of an earthquake on the Richter scale is given by $$ M=\frac{\log E-4.8}{1.5} $$ where \(E\) is the seismic wave energy in joules. According to this model, what is the magnitude of the demonstration quake? (It did not register above background noise overseas or on the seismograph of the Wolverton Seismic Vault, Hampshire. \()\)
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