Question

A hard rubber roller is cut with a ‘V-shaped’ profile at three locations as shown in the Top View in the image on the left. The roller is then dipped in blue colour and rolled over a white plane. Which option shows the correct impression on the plane?

• A. Fig A
• B. Fig B
• C. Fig C
• D. Fig D

Hint:

Any flat, angled cut on a cylinder always produces a curved, sinusoidal edge when the cylinder is unrolled. Sharp straight-edged diamonds only occur if the rolled object itself is flat.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question belongs to the topic of

Cylinder Unrolling and Surface Developments.
We are asked to determine the 2D pattern printed on a flat surface when a cylindrical rubber roller with three V-shaped notches is rolled over it.

Step 2: Key Formula or Approach:
To solve rolling cylinder problems:

• A flat planar cut made through a cylinder at an angle to its longitudinal axis does not unroll into straight lines.

• When unrolled, the intersection curve of a plane cutting a cylinder is mathematically represented by a portion of a sine wave (sinusoid).

• A V-shaped cut is composed of two intersecting planes, meaning the unrolled boundaries must be curved, meeting at sharp cusp-like vertices where the planes intersect.


Step 3: Detailed Explanation:

• Let us analyze the physical cuts on the roller:
- There are three locations with V-shaped notches.
- The unprinted (white) areas on the paper represent the regions where material has been removed from the roller's surface.
- Since the cuts are V-shaped, the boundaries of these unprinted shapes must show the unrolled profile of planar cuts.

• Let us evaluate the shape of these unprinted regions:
- If the cuts were straight when unrolled, the white shapes would be sharp, straight-sided diamonds. However, because the surface is cylindrical, the straight cuts warp into smooth curves upon unrolling. This eliminates Option (A).
- Let us examine Option (B): The shapes are horizontally elongated lens-like (lenticular) diamonds with smooth, curved boundaries that meet at sharp points on the top and bottom. This perfectly matches the unrolled geometry of two intersecting flat planar cuts.
- Options (C) and (D) show circular or hexagonal profiles, which would require hemispherical or polygonal cuts rather than flat V-shaped notches.


Step 4: Final Answer:
Option (B) represents the correct impression on the plane.