A rod of length L is positioned along the X-axis between x = 0 and x = L . The linear density (mass/length) of the rod varies with the distance x from the origin as = a + bx . Determine the mass of the rod in terms of a

Correct answer: A. aL + (bL^2)/(2) The mass of the rod is given by the integral of the linear mass density over its length. M = ∫_ 0 ^ L \, dx Substituting the given expression for : M = ∫_ 0 ^ L (a + bx) \, dx M = [ ax + (bx^2)/(2) ]_ 0 ^ L M = aL + (bL^2)/(2)

Concepts Of Physics MCQ Edition [Volume 1]

Exam: Concepts Of Physics MCQ Edition [Volume 1]
Subject: Physics
Chapter: Physics and Mathematics

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Physics and Mathematics

A rod of length L is positioned along the X-axis between x = 0 and...

A rod of length

L

is positioned along the X-axis between

x = 0

and

x = L

. The linear density (mass/length)

\rho

of the rod varies with the distance

x

from the origin as

\rho = a + bx

. Determine the mass of the rod in terms of

a

b

and

L

Options

A $aL + \dfrac{bL^2}{2}$
B $\dfrac{aL^2}{2} + \dfrac{bL^3}{3}$
C $aL + bL^2$
D $\dfrac{aL}{2} + \dfrac{bL^2}{3}$

Correct answer

A. $aL + \dfrac{bL^2}{2}$

Step-by-step solution

The mass of the rod is given by the integral of the linear mass density over its length.

M = \int_{0}^{L} \rho \, dx

Substituting the given expression for

\rho

M = \int_{0}^{L} (a + bx) \, dx

M = \left[ ax + \dfrac{bx^2}{2} \right]_{0}^{L}

M = aL + \dfrac{bL^2}{2}

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