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A thin rod length 'L' is lying along the x-axis with its ends at x=0 and x=L. Its linear density (mass/length) varies with x as k (x^(n))/(L), where n can be zero or any positive number. If the position x_(CM) of the centre of mass of the rod is plotted against 'n', which of the following graphs best approximates the dependence of x_(CM) on n? |
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Answer»
n is positive and as its value increases, the rate of increase of linear mass density with increase in x increases. This SHOWS that the centre of mass will SHIFT towards that end the rod where `n=L` as the value of n increases. Therefore graph (b) is ruled out. The linear mass density `lambda=k((x)/(L))^(n)` Here `(x)/(L)le1` With increase in the value of n the centre of mass shift towards the end `x=L` such that first the shifting is at a higher rate with increase in the value of n and then the rate decreases with teh value of n. These characteristics are represented by graph (a) `x_(CM)=(underset(0)overset(L)intx(lambdadx))/(underset(0)overset(L)intlambdadx)=(underset(0)overset(L)intx(lambdadx)^(n))/(underset(0)overset(L)intlambdadx)=(underset(0)overset(L)intk((x)/(L))^(n).xdx)/(underset(0)overset(L)intk((x)/(L))^(n)dx)` `(k[(x^(n+2))/((n+2)L^(n))]_(0)^(L))/([(kx^(n+1))/((n+1)L^(n))]_(0)^(L))=(L(n+1))/(n+2)` For `n=0,x_(CM)=(L)/(2),n=1,x_(CM)=(2L)/(3)`, `n=2,x_(CM)=(3L)/(4)` |
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