The formula for energy stored in the mechanical system of linear motion type is ______

Question

Examoogle · Accepted Answer

Option B: $\frac{1}{2}mv^2$

Explanation & Concepts:
In mechanical systems undergoing linear translational motion, energy is stored in the form of kinetic energy due to the mass of the body and its velocity. The formula for kinetic energy stored in a linear translational system is $\frac{1}{2}mv^2$, where $m$ is the mass in kilograms and $v$ is the linear velocity in meters per second.

Working Principle: Electromechanical energy conversion systems store energy in magnetic, electric, or mechanical forms. In linear mechanical systems, force applied to a mass causes linear acceleration, storing kinetic energy proportional to mass and the square of velocity.

Key Formulas: $E_{linear} = \frac{1}{2}mv^2$ — Kinetic energy stored in a linear motion mechanical system; $E_{rotational} = \frac{1}{2}J\omega^2$ — Kinetic energy stored in a rotational motion mechanical system; $E_{electric} = \frac{1}{2}CV^2$ — Energy stored in an electric field (Capacitor); $E_{magnetic} = \frac{1}{2}LI^2$ — Energy stored in a magnetic field (Inductor)

Key Points: Linear motion mechanical energy storage is analogous to electrical energy storage in an inductor or capacitor.; Mass $m$ in linear systems is analogous to moment of inertia $J$ in rotational systems.; Linear velocity $v$ is analogous to angular velocity $\omega$ in rotational systems.

Advantages: Simple linear relationship between kinetic energy and mass.; Direct analogical mapping between electrical and mechanical systems.

Disadvantages: Energy storage is limited by maximum allowable linear velocity and mechanical stress.; Linear systems often require stroke limits unlike continuous rotational systems.

Applications: Linear induction motors (LIM) and maglev trains.; Solenoid actuators and electromechanical relays.; Linear motion transducers and positioning systems.

Additional Info: Option A ($\frac{1}{2}J\omega^2$) represents energy stored in a rotational mechanical system.. Option C ($\frac{1}{2}CV^2$) represents energy stored in an electrostatic field of a capacitor.. Option D ($\frac{1}{2}LI^2$) represents energy stored in a magnetic field of an inductor.

Answer: B is correct — The formula for energy stored in a linear motion mechanical system is $\frac{1}{2}mv^2$, representing kinetic energy.

Exam Tip: Always remember the force-voltage and force-current analogies for GATE/ESE exams: Mass ($m$) corresponds to Inductance ($L$) in F-V analogy and Capacitance ($C$) in F-I analogy.

Examoogle · Answer

Option A: $\frac{1}{2}J\omega^2$

Examoogle · Answer

Option C: $\frac{1}{2}CV^2$

Examoogle · Answer

Option D: $\frac{1}{2}LI^2$

Feature	Linear System	Rotational System
Motion Type	Translational (Linear)	Rotational
Inertia Element	Mass ( $m$ )	Moment of Inertia ( $J$ )
Velocity Variable	Linear Velocity ( $v$ )	Angular Velocity ( $\omega$ )
Stored Energy Formula	$\frac{1}{2}mv^2$	$\frac{1}{2}J\omega^2$

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