Which of the options will result in the form shown on the left, when revolved around x-x' axis?
Show Hint
For solids of revolution problems, mentally "slice" the 3D object in half along its axis of rotation. The outline of this slice is the 2D profile you are looking for. Pay close attention to the details of corners and curves (e.g., V-groove vs. square groove).
Step 1: Understanding the Concept:
This question is about solids of revolution. A 3D shape is generated by rotating a 2D profile around an axis. We are given the 3D solid and need to identify the correct 2D profile that generates it when revolved around the x-x' axis.
Step 2: Key Formula or Approach:
The 2D profile represents the cross-section of the 3D object, taken from the axis of revolution outwards to the object's surface. We need to carefully trace the outline of the 3D object and match it to one of the given 2D profiles.
Step 3: Detailed Explanation:
Let's analyze the features of the 3D object from its central axis (x-x') outwards and from top to bottom.
At the very top, there is a cylindrical section of a certain diameter.
Below that, there is a smaller diameter cylindrical section, creating a step.
Below this step, there is a V-shaped groove, used for a V-belt in a pulley system.
Below the first groove, there is another cylindrical section.
Below this, there is a second, wider V-shaped groove.
Finally, at the bottom, there is the widest cylindrical section, which forms the base.
Now we compare this sequence of features with the profiles in the options:
(A) This profile shows angled surfaces instead of sharp V-grooves. Incorrect.
(B) This profile accurately shows all the features in the correct order: the top step, the first narrow V-groove, the middle cylindrical section, the second wider V-groove, and the wide base. This is a perfect match.
(C) This profile shows rectangular grooves, not V-shaped grooves. Incorrect.
(D) The features in this profile are in the wrong order and have the wrong shape. For example, it shows a groove at the very top. Incorrect.
Step 4: Final Answer:
By comparing the cross-sectional features of the 3D object with the given 2D profiles, we can conclude that option (B) is the correct profile.
