To find the normal stress \( \sigma \) at a location \( x \) along the length of a rod of length \( L \) under the influence of gravity, we start by considering the forces acting on the rod.
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Understanding the Forces: The weight of the portion of the rod below the location \( x \) will exert a downward force due to gravity. The weight of this portion can be expressed in terms of its volume and the density.
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Finding Volume and Weight:
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The volume \( V \) of the rod below location \( x \) is given by its cross-sectional area \( A \) multiplied by the length of the portion below \( x \) (which is \( L - x \)): \[ V = A * (L - x) \]
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The weight \( W \) of this portion is given by: \[ W = \rho * V * g \] Substituting for \( V \): \[ W = \rho * A * (L - x) * g \] Here, \( \rho \) is the density of the material, \( g \) is the gravitational acceleration, and \( g_0 \) can be used if it is specified that \( g \) is standard gravity.
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Finding the Normal Stress:
- The normal stress \( \sigma \) at location \( x \) is defined as the force over the area. The force acting at the location \( x \) is the weight of the portion of the rod below that point: \[ \sigma(x) = \frac{W}{A} \] Substituting for \( W \): \[ \sigma(x) = \frac{\rho * A * (L - x) * g}{A} \] The area \( A \) cancels out: \[ \sigma(x) = \rho * (L - x) * g \]
Thus, the expression for the normal stress \( \sigma \) at location \( x \) as a function of \( \rho \), \( L \), and \( g \) is:
\[ \sigma(x) = \rho * (L - x) * g \]