1. Question 1

A cylinder with a variable cross section radius is considered as shown in Figure 1. The radius of the cylinder varies linearly from r1 at the clamped end to r2 at the other end. The length of the cylinder is L. A torque T is applied at the free end. How much is the rotation of the cross section at the free end?

This chunk outlines the problem we're trying to solve. We have a cylinder that is clamped at one end and free at the other, with a torque applied. The radius of this cylinder changes gradually along its length. The goal is to determine how much the free end rotates when the torque is applied.

Think of this scenario like twisting a soft rubbery toy at one end while holding the other. If the toy's thickness varies from top to bottom, the amount it twists will depend on how thick it is at any point—just like how the varied radius of the cylinder influences its rotation under torque.

In the previous lecture, we had derived that the end-to-end rotation Ω is given by equation (1), but it holds only when the cross-section is uniform over the entire length. 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 this portion, we recognize that the simple formula for rotation applies only to uniform cylinders. When the cylinder's radius varies, we need to use a different formula that relates the twist across the length of the cylinder. The twist, described by κ, reflects how the cylinder rotates differently at each point along its length.

Picture a spiral staircase. As you move up the stairs, the twists become steeper or shallower depending on the design. Similarly, the twist in our cylinder varies, showing how different sections rotate differently based on their geometry.

The variation of T along the axis of the cylinder can be found by making a free body diagram. We cut a section of the cylinder at a distance x from the clamped end. The external torque T acts on the right end of the right section.

Here, we introduce free body diagrams, which help visualize the forces acting on the cylinder. By 'cutting' a section of the cylinder, we can analyze the torques that act at different points. This approach will help us understand how the torque is constant even if the radius changes.

Think of a long, flexible stick. If you apply a force at one end, the tension and torque applied are distributed throughout the stick's length, even if the stick bends. This is similar to how the torque behaves in our cylinder.

This means that the torque at every cross-section is the same and equal to the applied external torque T even though cross-sectional radius is changing. Assuming that the cylinder is made up of a single material, its shear modulus will also be constant, i.e., G(x) = G.

In this chunk, we learn that despite changing cross-sectional areas, the torque remains consistent throughout the length of the cylinder. Additionally, if the cylinder is homogenous (made of the same material), its shear modulus—resistance to deformation—also remains unchanged. This consistency is crucial for our calculations.

Consider a large rubber band stretched evenly across its length. Even if some parts of the band might look thinner than others, the force applied on one end is felt uniformly because the material properties are the same throughout.

Using equation (4), the end-to-end rotation can be found by integrating twist along the axis of the cylinder, i.e., (9). As r varies linearly along the axis of the cylinder and r(0)=r1, r(L)=r2, (10).

To compute the total rotation of the cylinder from one end to the other, we need to integrate the twist we calculated over the length of the cylinder. This is where we apply the linear variation of radius to derive the required total rotation across the entire length of the cylinder.

Imagine filling a long tube with different amounts of colored sand at each point; if you then twist the tube, the pattern of colors along the tube will help visualize how each section contributes to the total twist, just as the varying radius contributes to the total rotation in our cylinder.

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