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Chapter 4: Problem 28

A particle is acted upon by a force \(F=k x,(k>0)\), where \(x\) is displacement of particle. If potential energy at origin is zero, then the potential energy of the particle varies with \(x\) as

Short Answer

Expert verified
The potential energy of the particle varies with displacement \(x\) as \(U(x) = - \frac{1}{2}kx^2\).

Step by step solution

01

Express force as derivative of potential energy

In physics, force is obtained from the negative gradient of potential energy. Mathematically, it is represented as \(F = - \frac{dU}{dx}\), where \(U\) is the potential energy and \(x\) is the displacement.
02

Integrate to find potential energy

To obtain the potential energy, we can integrate both sides of this equation with respect to \(x\). The integral of force \(F\) with respect to \(x\) will provide us the equation for potential energy \(U\).
03

Substitution and Integration

Substituting the given force \(F = kx\) into \(F = - \frac{dU}{dx}\), we get the differential equation: \(-kx = \frac{dU}{dx}\). We can solve this differential equation by integrating both sides with respect to \(x\). This gives us the equation for potential energy \(U\) as a function of displacement \(x\). \(U(x) = - \frac{1}{2}kx^2\).
04

Verification with boundary conditions

Upon verifying the boundary condition that potential energy at the origin \(x=0\) is zero, we can see that this equation holds: \(U(0) = - \frac{1}{2}k*0^2 = 0\)

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Most popular questions from this chapter

With what minimum speed \(v\) must a small ball should be pushed inside a smooth vertical tube from a height \(h\) so that it may reach the top of the tube? Radius of the tube is \(R\). (Assume radius of cross-section of tube is negligible in comparison to \(R\).) (A) \(\sqrt{2 g(h+2 R)}\) (B) \(\frac{5}{2} R\) (C) \(\sqrt{g(5 R-2 h)}\) (D) \(\sqrt{2 g(2 R-h)}\)

When a body moves in a circle, the work done by the centripetal force is always \((\mathrm{A})>0\) (B) \(<0\) (C) Zero (D) None of these

A body constrained to move in \(y\)-direction is subjected to a force given by \(\vec{F}=(-2 \vec{i}+15 \vec{j}+6 \vec{k}) N\). The work done by this force in moving the body a distance of \(10 \mathrm{~m}\) along the \(y\)-axis is (A) \(20 \mathrm{~J}\) (B) \(150 \mathrm{~J}\) (C) \(60 \mathrm{~J}\) (D) \(190 \mathrm{~J}\)

A mass \(m=1 \mathrm{~kg}\) moving horizontally with velocity \(v_{0}=2 \mathrm{~m} / \mathrm{s}\) collides in elastically with a pendulum of same mass. Find the maximum change in potential energy (in Joule) of combined mass.

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