Question:
A particle, which is constrained to move along x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F (x) $=-kx+ax^3$. Here, k and a are positive constants. For $x \ge 0$, the functional form of the potential energy U (x) of the particle is
A particle, which is constrained to move along x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F (x) $=-kx+ax^3$. Here, k and a are positive constants. For $x \ge 0$, the functional form of the potential energy U (x) of the particle is
Updated On: Jun 14, 2022
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The Correct Option is D
Solution and Explanation
$F=-\frac{dU}{dx}$
$\therefore \hspace10mm dU=-F.dx \, or \, U(x)=-\int_0^x(-kx+ax^3)dx$
$\hspace10mm U(x)=\frac{kx^2}{2}-\frac{ax^4}{4}$
$\hspace10mm U(x)=0 \, at \, x=0 \, and \, x=\sqrt{\frac{2k}{a}}$
$\hspace10mm U(x)=negative \, for \, x=\sqrt{\frac{2k}{a}}$
From the given function, we can see that
F = O at x = 0 i.e. slope of U-x graph is zero at
x = 0. Therefore, the most appropriate option is (d).
$\therefore \hspace10mm dU=-F.dx \, or \, U(x)=-\int_0^x(-kx+ax^3)dx$
$\hspace10mm U(x)=\frac{kx^2}{2}-\frac{ax^4}{4}$
$\hspace10mm U(x)=0 \, at \, x=0 \, and \, x=\sqrt{\frac{2k}{a}}$
$\hspace10mm U(x)=negative \, for \, x=\sqrt{\frac{2k}{a}}$
From the given function, we can see that
F = O at x = 0 i.e. slope of U-x graph is zero at
x = 0. Therefore, the most appropriate option is (d).
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Concepts Used:
Work, Energy and Power
Work:
- Work is correlated to force and the displacement over which it acts. When an object is replaced parallel to the force's line of action, it is thought to be doing work. It is a force-driven action that includes movement in the force's direction.
- The work done by the force is described to be the product of the elements of the force in the direction of the displacement and the magnitude of this displacement.
Energy:
- A body's energy is its potential to do tasks. Anything that has the capability to work is said to have energy. The unit of energy is the same as the unit of work, i.e., the Joule.
- There are two types of mechanical energy such as; Kinetic and potential energy.
Read More: Work and Energy
Power:
- Power is the rate at which energy is transferred, conveyed, or converted or the rate of doing work. Technologically, it is the amount of work done per unit of time. The SI unit of power is Watt (W) which is joules per second (J/s). Sometimes the power of motor vehicles and other machines is demonstrated in terms of Horsepower (hp), which is roughly equal to 745.7 watts.
- Power is a scalar quantity, which gives us a quantity or amount of energy consumed per unit of time but with no manifestation of direction.