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Torque is a measure of the rotational force applied to an object, causing it to rotate around an axis. In the case of cylinders, understanding torque is crucial because it directly affects how the material twists under the applied force. Torque is represented by the symbol \( T \) and is measured in units like Newton-meters (Nm). When dealing with rotational systems like cylinders, torque is crucial because:

• It determines the amount of force applied to rotate the cylinder.
• In a system where multiple cylinders are joined, like in the exercise, torque remains constant across the system if they are in series.

In the problem, since both cylinders A and B experience the same torque, it is pivotal to understand that while the force stays the same, it is distributed differently across each cylinder due to their differing geometry, particularly their radii ratio of 1:2.

The polar moment of inertia is an important geometric property when assessing how objects resist torsion, or twisting. For a cylinder, the polar moment of inertia \( J \) is expressed through the formula \( J = \frac{\pi r^4}{2} \), where \( r \) is the radius of the cylinder.In our problem:

• Cylinder A has radius \( r \) and a polar moment of inertia \( \frac{\pi r^4}{2} \).
• Cylinder B, being of twice the radius (2r), has a significantly larger polar moment of inertia \( 16 \times \frac{\pi r^4}{2} \).

The polar moment of inertia indicates how strongly a cross-sectional area of a cylinder resists twisting. Therefore, cylinder B, having a much larger polar moment of inertia compared to cylinder A, resists twisting more effectively. This contrast affects the angle of twist each cylinder experiences, as a smaller \( J \) means a greater angle of twist for the same torque.

Shear modulus, often denoted as \( G \), measures a material's ability to resist deformation under shear stress. It is a fundamental property of materials that describes their elasticity in response to applied torque. The higher the shear modulus, the more resistant a material is to deformation. The units are typically Pascals (Pa).In the context of rotating cylinders subjected to torque:

• Shear modulus contributes to the formula for angle of twist: \( \phi = \frac{TL}{JG} \).
• Both cylinders A and B are made of the same material, meaning they share the same shear modulus \( G \).

This uniformity in shear modulus across both cylinders implies that any difference in the angle of twist is solely due to variations in their geometry (radii) and not because of the material's resistance to deformation. Understanding shear modulus helps predict how a given material will react under specific forces, which is crucial for calculating how much a cylinder will twist when torque is applied.