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Chapter 5: Problem 55

A particle moves in a straight line with retardation proportional to its displacement. kinetic energy during a displacement \(x\) is proportional to (a) \(x^{2}\) (b) \(e^{x}\) (c) \(x\) (d) \(\log _{e} x\)

Short Answer

Expert verified
The kinetic energy is proportional to \(x^2\), option (a).

Step by step solution

01

Understanding the Problem

We are given that a particle moves with retardation (negative acceleration) proportional to its displacement. This implies that the force acting on it is proportional to the displacement, meaning that the restoring force is like a spring force. We need to determine how the kinetic energy varies with displacement.
02

Expressing Retardation Mathematically

Given that the retardation is proportional to displacement, we write: \[ a = -kx \]where \( a \) is the acceleration, \( x \) is the displacement, and \( k \) is a positive constant of proportionality.
03

Using Newton's Second Law

From Newton鈥檚 second law, the force \( F \) is given by the mass \( m \) times acceleration \( a \). Therefore:\[ F = ma = -mkx \]This shows the force is like a spring force, with spring constant \( mk \).
04

Understanding Kinetic Energy

Kinetic energy \( K \) is given by the formula: \[ K = \frac{1}{2}mv^2 \]Using energy conservation in a harmonic motion setting or deriving from dynamics, for small displacement, at maximum displacement (where velocity is zero), potential energy is maximum and correlated to displacement. The particle's maximum kinetic energy corresponds to when potential energy is at this reference.
05

Relating to Harmonic Motion

Since the situation is similar to a simple harmonic oscillator, we find that energy trade-off follows:Total energy, i.e., sum of kinetic and potential energy, in cases of simple harmonic motion, provides forms: \[ E = \frac{1}{2}kx^2 = K + U \] Thus, kinetic energy should vary similarly. Recognizing from reference energy states, one concludes kinetic energy correlation to square of displacement based on oscillation principles.
06

Selecting the Correct Option

Given the relation that the system resembles harmonic motion, the kinetic energy should be proportional to displacement squared when considering about energy forms traded within positions.Thus, the option fitting well within natural motion equations is: (a) \(x^2\)

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Key Concepts

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

Simple Harmonic Motion
Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position and acts in the direction opposite to the displacement. This kind of motion is typical for systems like pendulums and springs.
In SHM, the motion repeats itself at regular intervals. The essential characteristics are:
  • Motion is central and oscillates back and forth over a certain path.
  • Restoring force follows Hooke's law, being proportional and opposite to displacement.
  • Acceleration is also opposite and proportional to displacement.
The goal in understanding SHM is to recognize how forces and energies behave over time. Here, the particle's motion as given implies a scenario where the velocity and kinetic energy vary regularly with displacement, akin to SHM.
Retardation Proportional to Displacement
The concept of retardation proportional to displacement implies that as a particle moves, its acceleration decreases linearly with its position. This type of behavior is indicative of a negative restoring force, much like a spring pulling back to its natural length.
Mathematically, this can be expressed as:
  • Acceleration \( a = -kx \) where \( x \) is displacement and \( k \) is a constant.
  • The negative sign indicates that the acceleration opposes the direction of the displacement.
  • This relationship is key to understanding both motion and energy changes.
This basic model provides an easy way to assess how forces in a retarding system might behave, offering insights into potential and kinetic energy distribution.
Newton's Second Law
Newton's second law of motion forms the foundation for analyzing forces acting on a particle. It states that the force applied to an object is equal to its mass multiplied by its acceleration (\( F = ma \)). This fundamental principle provides insight into how an object will move under various forces.
For the given scenario, the application of Newton's second law leads to:
  • Using \( F = ma \) and substituting the acceleration \( a = -kx \), resulting in \( F = -mkx \).
  • Reveals that the force is equivalent to that seen in spring systems, providing consistency with harmonic motion behavior.
This framework is crucial for using mathematics to predict movements, which is essential for understanding energy transitions.
Energy Conservation in Mechanics
Energy conservation is a key concept in mechanics, stating that the total energy within a closed system remains constant. In the case of mechanics involving motion, this typically involves potential and kinetic energy transformations.
Understanding energy distribution requires:
  • Identifying how potential energy builds up when kinetic energy decreases, and vice versa.
  • Recognizing that in simple harmonic motion, total mechanical energy is \( E = K + U \) where \( K \) is kinetic energy and \( U \) is potential energy.
  • For proportional retardation scenarios, kinetic energy evolves from the potential energy given by \( \frac{1}{2} kx^2 \).
In such systems, recognizing that kinetic energy is proportional to displacement squared (\( x^2 \)) is important, shedding light on motion characteristics and affirming conservation principles.

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Most popular questions from this chapter

Consider a hypothetical relation giving potential energy of a system of two atoms in a diatomic molecule: \(U-\frac{\alpha}{r^{11}} \frac{\beta}{r^{5}}\), (where \(\alpha\) and \(\beta\) are constant, \(r\) represents interatomic scparation). Interatomic separation at the equilibrium is (a) \(\left(\frac{11 \alpha}{5 \beta}\right)^{1 / 6}\) (b) \(\left(\frac{11 \alpha}{\beta}\right)^{1 / 6}\) (c) \(\left(\frac{5 \beta}{11 \alpha}\right)^{1 / 6}\) (d) \(\left(\frac{\beta}{11 \alpha}\right)^{1 / 6}\)

A uniform ring, having radius \(a\) and mass \(m\) is to be rotated in the horizontal plane about its own axis with constant angular velocity \(\omega\), what would be the tension in the ring and nature of force? (a) \(\frac{m a \omega^{2}}{2 \pi}\) tensile (b) \(m a \omega^{2}\) tensile (c) \(\frac{m a \omega^{2}}{2}\) compressive (d) \(m a \omega^{2}\) compressive

Potential energy of a particle moving along \(x\) -axis is given by \(U=(a x-b) x\). Speed of the particle is maximum at \(x\) equal to (here \(a\) and \(b\) are positive constants) (a) Zero (b) \(\frac{b}{2 a}\) (c) \(\frac{b}{a}\) (d) \(\frac{2 b}{a}\)

\(\Lambda\) chain of mass \(m\) and length \(l\) is held vertical, such that its lower end just touches the floor. I released from rest. Find the force exeried by the chain on the table when upper end is about to hit the foor. Solution Force \(F\) exerted by chain consists of two components (a) \(F_{1}\) weight of the fallen portion of the chain, (b) \(F_{2}\) thrust of the falling part of chain. Now consider an clement of chain of length \(d y\) at a height \(y\) from the floor. It will strike the floor with a velocity \(v-\sqrt{2 g y}\). Thus we have, \(\Gamma_{1}=\lambda y g\) Here \(\lambda\) is the mass per unit length of chain and \(\Gamma_{2}-v_{\mathrm{rel}} \frac{d m}{d t}\) We have \(v_{\mathrm{rel}}=v \quad\) and \(\quad d m=\lambda d x \quad \therefore F=-v \frac{\lambda d x}{d t}-\lambda v^{2}\) 'Ihe force exerted by chain on the floor,$$ F=F_{1}+F_{2}=\lambda y g+\lambda v^{2}-\lambda y g+\lambda(\sqrt{2 g y})^{2}=\lambda y g+2 \lambda g=3 \lambda y g $$ When upper end is about to hit the floor, \(y=l\) \(\therefore \quad F=3 \lambda / g=3 m g\)

Statement-1: A body is moving under the action of constant force, Its kinetic energy observed from a uniformly moving frame along any fixed direction changes. Statement-2: The acceleration of a particle observed from a uniformly moving frame is independent on the frame of references.
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