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Chapter 7: Problem 35

At which point in the swing of an ideal pendulum (ignoring friction) is the gravitational potential energy at its maximum? At which point is the kinetic energy at its maximum?

Short Answer

Expert verified
Potential energy is maximum at the highest points; kinetic energy is maximum at the lowest point.

Step by step solution

01

- Understanding Energy in a Pendulum

In an ideal pendulum, energy is conserved and oscillates between gravitational potential energy and kinetic energy. It is crucial to identify the key points in the swing where these energies reach their maximum values.
02

- Gravitational Potential Energy

Gravitational potential energy is at its maximum when the pendulum is at the highest point in its swing. This is because the height relative to the lowest point (where the kinetic energy would be maximum) is greatest here.
03

- Kinetic Energy

Kinetic energy is at its maximum at the lowest point in the pendulum's swing. At this point, the potential energy is at its minimum, meaning all the energy has been converted into kinetic energy.
04

- Conclusion

To summarize: The gravitational potential energy is maximum at the highest points in the swing, and the kinetic energy is maximum at the lowest point in the swing.

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Key Concepts

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

Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy stored in an object due to its position in a gravitational field. For a pendulum, this energy is maximum at the highest points in its swing. Here, the pendulum is farthest from its lowest position, meaning it has the most height and thus the most potential energy. Mathematically, gravitational potential energy can be expressed as: \[ GPE = mgh \] where \( m \) is the mass of the pendulum bob, \( g \) is the acceleration due to gravity, and \( h \) is the height above the lowest point. When the pendulum reaches its peak on either side, all of the energy is stored as gravitational potential energy and its speed is momentarily zero.
Kinetic Energy
Kinetic energy (KE) is the energy of motion. For the pendulum, kinetic energy is maximum when the pendulum is at the lowest point in its swing. At this point, the pendulum is moving the fastest and all the energy that was stored as potential energy at the highest points is converted to kinetic energy. The formula for kinetic energy is: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of the pendulum bob and \( v \) is its velocity. Thus, at the lowest point, potential energy is at its minimum and kinetic energy is at its maximum. As the pendulum rises again, this kinetic energy will be transformed back into potential energy.
Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of a pendulum:
  • At the highest points in the swing, all the energy is gravitational potential energy.
  • At the lowest point, all of that potential energy is converted to kinetic energy.
  • Through the swing, energy oscillates between potential and kinetic forms.
This ensures that the total mechanical energy of the system (potential + kinetic energy) remains constant, assuming no energy loss due to friction or air resistance.
Pendulum Motion
A pendulum consists of a mass (known as the bob) suspended from a pivot so that it can swing freely. The motion of a pendulum can be described by several key characteristics:
  • At the highest points in its swing, the bob's gravitational potential energy is at a maximum and its kinetic energy is at a minimum.
  • At the lowest point, the bob's kinetic energy is at its maximum while gravitational potential energy is at its minimum.
  • The pendulum moves fastest at the lowest point and slowest at the highest points.
  • The motion is ideally periodic, meaning it repeats in regular intervals.
Understanding pendulum motion is fundamental to many areas of physics, including the study of wave motion and harmonic oscillators.

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Most popular questions from this chapter

A \(20-\mathrm{N}\) block lifted straight upward by a hand applying a force of \(20 \mathrm{N}\) has an initial kinetic energy of \(16 \mathrm{J} .\) If the block is lifted \(1 \mathrm{m},\) how much work does the hand do? What is the block's final kinetic energy?

A 2 -kg block is released from rest at the top of a 20 -mlong frictionless ramp that is \(4 \mathrm{m}\) high. At the same time, an identical block is released next to the ramp so that it drops straight down the same \(4 \mathrm{m}\). What are the values for each of the following for the blocks just before they reach ground level? Which quantities are the same for the two blocks? a. gravitational potential energy b. kinetic energy c. speed d. momentum

You reach out a second-story window that is 5 m above the sidewalk and throw a 0.1 -kg ball straight upward with \(6 \mathrm{J}\) of kinetic energy. a. What is the ball's gravitational potential energy when it is released? b. What is the ball's gravitational potential energy just before hitting the sidewalk? c. What is the ball's kinetic energy just before hitting the sidewalk? d. How would the answer to part (c) change if the ball had initially been thrown straight down with \(6 \mathrm{J}\) of kinetic energy?

Two cars have different masses but the same kinetic energies. If the same frictional force is used to stop each car, which car, if either, will stop in the shorter distance?

Two identical cars traveling at the same speed collide headon and come to rest in a mangled heap. At first glance it appears that energy is not conserved in this collision. However, like Dennis's mother in Richard Feynman's story at the beginning of the chapter, we find the energy "hidden" in many different forms. The initial kinetic energy is transformed into sound energy, thermal energy, and deformation energy. Where does the initial momentum of the system hide?
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