Below you see a picture of a rod of length , suspended from a ceiling high up.

Because of the force of gravity, the material in the rod experiences a normal stress. Assuming a constant gravitational acceleration , find an expression for the normal stress at location as a function of ,, and .

Explicitly indicate multiplication with a * symbol, indicate division with a / symbol and write greek letters out in full (so is written as "rho"). Additionally, denote subscripts using an underscore (so is written as "g_0").

1 answer

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.

  1. 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.

  2. Finding Volume and Weight:

    • 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 \( L - x \)): \[ V = A * (L - x) \]

    • 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.

  3. 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:

\[ \sigma(x) = \rho * (L - x) * g \]