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JEE Main 2008
Exam
JEE Main
Year
2008
Subject
Physics
Chapter
Center of Mass, Momentum and Collision

JEE Main 2008 · Physics · Center of Mass, Momentum and Collision

A thin rod of length ' L ' is lying along the x -axis with its ends...

A thin rod of length ' LL ' is lying along the xx-axis with its ends at x=0andx=Lx=0 and x=L. Its linear density (mass/length) varies with xx ask \left(\frac{x}{L}\right)^n,where, wherencanbezerooranypositivenumber.Ifthepositioncan be zero or any positive number. If the positionx_{\mathrm{CM}}ofthecentreofmassoftherodisplottedagainst′of the centre of mass of the rod is plotted against 'n′,whichofthefollowinggraphsbestapproximatesthedependenceof', which of the following graphs best approximates the dependence ofx_{\mathrm{CM}}ononn$ ?A thin rod of length ' L ' is lying along the x -axis with its ends at x=0 and x=L . Its linear density (mass/length)A thin rod of length ' L ' is lying along the x -axis with its ends at x=0 and x=L . Its linear density (mass/length)A thin rod of length ' L ' is lying along the x -axis with its ends at x=0 and x=L . Its linear density (mass/length)A thin rod of length ' L ' is lying along the x -axis with its ends at x=0 and x=L . Its linear density (mass/length)

Options

  1. AWhich of the following is correct with respect to the following statements? Due to diffusion of electrons from n to p
  2. BWhich of the following is correct with respect to the following statements? Due to diffusion of electrons from n to p
  3. CWhich of the following is correct with respect to the following statements? Due to diffusion of electrons from n to p
  4. DWhich of the following is correct with respect to the following statements? Due to diffusion of electrons from n to p

Correct answer

A. Which of the following is correct with respect to the following statements? Due to diffusion of electrons from n to p

Step-by-step solution

xcm=∫dmx∫dm=∫λdx⋅x∫dm=∫k(xL)n⋅xdx∫k(xL)ndx=[kxn+2(n+2)Lnkxn+1(n+1)Ln]0L=[x(n+1)n+2]0Lxcm=L2,2L3,3L4,4L5,5L6,…\begin{aligned}& x_{c m}=\frac{\int d m x}{\int d m}=\frac{\int \lambda d x \cdot x}{\int d m}=\frac{\int k\left(\frac{x}{L}\right)^n \cdot x d x}{\int k\left(\frac{x}{L}\right)^n d x}=\left[\frac{\frac{k x^{n+2}}{(n+2) L^n}}{\frac{k x^{n+1}}{(n+1) L^n}}\right]_0^L=\left[\frac{x(n+1)}{n+2}\right]_0^L \\ & x_{c m}=\frac{L}{2}, \frac{2 L}{3}, \frac{3 L}{4}, \frac{4 L}{5}, \frac{5 L}{6}, \ldots \end{aligned}$$
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