2.1. Solution

In the previous lecture, we had derived that the end-to-end rotation Ω is given by (1) but it holds only when the cross-section is uniform over the entire length.

In the earlier discussion, we established an equation that relates to the end-to-end rotation of a cylinder. This equation is applicable when the cross-section of the cylinder remains consistent from one end to the other, meaning that its diameter doesn't change. Thus, uniformity in cross-section is critical for this simple relationship to hold true.

Think of a uniform spaghetti strand versus a string of different lengths. If you twist one end of the uniform spaghetti, the entire strand rotates uniformly. However, if a string has varying thickness, twisting it isn't as straightforward — some parts may twist more, while others twist less.

For the current case, a different form of the above equation is useful as shown below: (2) twist(κ) where κ denotes twist or the rate of change of rotation of the cross section.

In contrast to the uniform cross-section, when dealing with a cylinder that has a variable cross-section, we need a new equation to describe its behavior. Here, the 'twist' (κ) refers to how much the cross-section twists per unit length along the cylinder. This twist can change from one part to another due to the varying radius.

Imagine twisting a rubber band with varying thickness; thicker parts might resist twisting more than thinner ones. Similarly, in a cylinder with varying radii, different sections respond uniquely to the applied torque, which is captured by the variable twist.

As the same torque T acts in every cross-section (as shown below), the above equation implies that twist varies along the length as the polar moment of area J is varying here due to varying radius.

When a torque (T) is applied, it exerts a twisting force along the entire length of the cylinder. However, as the radius varies from one section to another, the way the torque affects twisting changes too. This variability in cross-section translates into different values for the polar moment of area (J), which is an important factor in calculating the twist in each segment of the cylinder.

Consider a spiral staircase where each step varies in width. Pushing down on the center affects how each step turns. In a cylinder, the same pushes (torque) affect sections differently because of their widths (radii).

To find T(x) for a cross-section of radius r(x), we integrate over the area of the cross section, i.e., (7)

To determine the torque at any specific point along the cylinder (denoted as T(x)), we look at how the radius changes and calculate the effects over the area of each cross-section. Integrating over the area helps us take into account varying widths at each position, giving a comprehensive picture of how torque distributes within the material.

Imagine trying to lift a seesaw at different points. By calculating the force needed for each point (like the area of a cross-section), you can determine how much torque to apply to lift it seamlessly.

The variation of T along the axis of the cylinder can be found by making a free body diagram.

A free body diagram is a visual representation that helps in understanding the forces acting on the cylinder. By segmenting the cylinder and analyzing the forces acting at each cut section, we can set up equations based on torque balance. This understanding is crucial for determining how torque varies along the length and ensuring that all forces are accounted for.

Think of a tightrope walker balancing at a point — if they were to draw lines representing the forces acting on them, they'd better decipher how to maintain equilibrium. Similarly, for the cylinder, drawing forces helps visualize how to achieve balance.

Assuming that the cylinder is made up of a single material, its shear modulus will also be constant, i.e., G(x) = G.

If we make an assumption that the entire cylinder is made from one uniform material, the shear modulus (G), which measures the material's resistance to shear deformation, remains constant along its length. This simplification allows for easier calculations as it reduces the variability involved in determining the twisting response of the cylinder.

Like a uniform piece of clay, if you apply pressure to twist it, the entire mass behaves similarly. If the clay were mixed with different materials, its resistance to your twist might vary in different sections.

--