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Chapter 5: Problem 14

Statement-1: If the work done by forces other than conservative forces is zero, then mechanical energy is conserved. Statement-2: The work done by non- conservative forces is always negative.

Short Answer

Expert verified
Statement-1 is true; Statement-2 is false.

Step by step solution

01

Understanding Statement-1

Statement-1 claims that if the work done by non-conservative forces is zero, then the mechanical energy of a system remains constant or conserved. This is a true statement because mechanical energy (sum of kinetic and potential energy) is conserved when no net work is done by non-conservative forces, adhering to the principle of conservation of energy.
02

Analyzing Statement-2

Statement-2 suggests that the work done by non-conservative forces is always negative. This statement is false. Non-conservative forces, such as friction or air resistance, usually do negative work when they oppose the motion. However, if they act in the direction of motion, their work can be positive.
03

Evaluating the Statements

Now, compare both statements. Statement-1 is correct as it accurately describes a scenario where mechanical energy remains constant when no work is done by non-conservative forces. Statement-2, however, is incorrect because the work done by non-conservative forces is not always negative.

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Key Concepts

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

Conservative Forces
Conservative forces are those that conserve mechanical energy within a system. Common examples include gravitational and elastic forces. When work is done by conservative forces, the energy can be completely converted from potential to kinetic energy or vice versa, without any loss to the surroundings.
Consider gravity; when an object falls, the gravitational force does work on the object, converting potential energy to kinetic energy. Upon reaching the ground, all potential energy may transform into kinetic.
  • The work done by conservative forces depends only on the initial and final positions, not the path taken.
  • Conservative forces allow a system to regain its original energy state after one complete cycle (like a pendulum).
  • The potential energy associated with conservative forces can be stored or retrieved without energy loss.
Understanding these forces help determine when the conservation of mechanical energy applies to simplify solving physics problems.
Non-conservative Forces
Non-conservative forces differ from conservative forces as they do not conserve mechanical energy. Examples include friction and air resistance. These forces dissipate energy in forms such as heat, sound, or deformation, making it irretrievable in terms of mechanical energy within the system.
  • When these forces act, they often do negative work, opposing the motion. However, non-conservative work can occasionally be positive if they assist a movement.
  • The work done depends on the path taken, not just the initial and final positions.
  • Because of energy loss, systems influenced by non-conservative forces require external energy sources to maintain motion (like a car engine overpowering friction).
Mastering this concept is crucial, especially when analyzing energy losses in real-world scenarios, where such forces are prevalent.
Work and Energy
Work and energy are closely related in physics, tied together by the work-energy principle. Work is done when a force moves an object over a distance, and this work can lead to a change in the object's energy. There are different types of energy involved, specifically kinetic energy and potential energy, which make up the system's total mechanical energy.
  • Work is calculated using the formula: \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the force, \( d \) is the distance object moves, and \( \theta \) is the angle between the force and the motion direction.
  • Mechanical energy is the sum of kinetic energy (energy of motion) and potential energy (stored energy).
  • The work-energy principle states that the work done on a system is equal to the change in its kinetic energy: \( W = \Delta KE \).
Understanding work and energy provides a foundation for analyzing physical interactions, affirming whether a system's mechanical energy is conserved or altered by various forces.

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Most popular questions from this chapter

Two particles of masses \(m_{1}\) and \(m_{2}\) are connected by an inextensible string which passes through a smooth vertical tube. The particle \(m_{1}\) is in equilibrium by the tension in the string which revolves the particle \(m_{2}\) in a horizontal circular path. Find the (a) angle \(\theta\) and (b) angular speed \(\omega\).

Work done in time \(t\) on a body of mass \(m\) which is accelerated from rest to a speed \(v\) in time \(t_{1}\), as a function of time \(t\) is given by (a) \(\frac{1}{2} m \frac{v}{t_{1}} t^{2}\) (b) \(m \frac{v}{t_{1}} t^{2}\) (c) \(\frac{1}{2} \frac{m v^{2}}{t_{1}} t^{2}\) (d) \(\frac{1}{2} m \frac{v}{t_{1}} t^{2}\)

A conscrvative force ficld is given by \(F-\frac{k}{r^{2}}\), where \(k\) is a constant assuming zero potential energy at \(r=r_{0}\), Then (a) Potential energy \(U(r)=\frac{k}{r}-\frac{k}{r_{e}}\) (b) Potential energy at \(r=\infty, U=\frac{-k}{r_{0}}\) (c) Potential encrgy at \(r=x\) is minimum (d) Nonc

Which of the following force is/are not conservative? (a) \(\vec{F}-a \hat{i}+b \hat{j}\) (b) \(\vec{F}-a x \hat{i}+b y \hat{j}(a \neq b)\) (c) \(\vec{F}-x y \hat{i}\) (d) \(\vec{F}-x^{2} y \hat{i}-2 x y^{2} \hat{j}\) (e) \(\vec{F}-\left(a x+b x^{3}+c x^{4}\right) \hat{i}\) (f) \(\vec{F}-A x^{2} \hat{i}+B x y \hat{j}\)

If the kinctic energy of a particle continuously increases with time, then (a) the magnitude of its lincar momentum also incrcases continuously with time. (b) the height above the ground level must continuously decrcase. (c) the resultant force on the particle must be parallel to the velocity al all instants. (d) the resultant force on the particle must be at an angle less than \(90^{\circ}\) all the time.
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