Question

A clock is displaced horizontally at a uniform rate along the direction shown. If the tip of the hour hand is traced from 2 AM to 9 AM, which option shows the correct tracing pattern?

• A. A
• B. B
• C. C
• D. D

Hint:

For problems involving combined motion, break down the movement into its components (e.g., horizontal and vertical). Analyze each component separately over the given interval and then synthesize them to visualize the final path.

Answer 1

The Correct Option is A

Step 1: Understanding the Motion:
The path of the hour hand's tip is a result of two simultaneous motions:
1. Rotational Motion: The tip of the hour hand moves in a circle, completing one full rotation in 12 hours.
2. Translational Motion: The entire clock moves horizontally to the right at a constant speed.
Step 2: Analyzing the Motion from 2 AM to 9 AM:
The total time elapsed is 7 hours.
- Rotational Path: At 2 AM, the hour hand is in the upper-right quadrant. As it moves towards 9 AM, it first rotates downwards, passing its lowest point at 6 AM. After 6 AM, it begins to move upwards, reaching the 9 AM position in the upper-left quadrant. Therefore, the vertical component of the motion is first downwards, then upwards.
- Translational Path: Throughout this period, the clock is continuously moving to the right. So, the horizontal position of the tip is always increasing.
Step 3: Combining the Motions:
The combination of these two motions will produce a curve known as a trochoid. The curve must show a continuous progression to the right. Vertically, it should start at a high point (corresponding to 2 AM), dip to a minimum point (corresponding to 6 AM), and then rise again to an intermediate high point (corresponding to 9 AM).
Step 4: Evaluating the Options:
- Option A: This curve correctly shows a continuous movement to the right, while starting high, dipping to a low point, and then rising again. This perfectly represents the combined motion.
- Option B: This curve only shows a downward vertical motion, which is incorrect as the hand moves up after 6 AM.
- Option C: This curve only shows an upward vertical motion, which is incorrect as the hand moves down from 2 AM to 6 AM.
- Option D: This shows a symmetrical wave, but the motion of the hour hand is not symmetrical over this period (4 hours down, 3 hours up). The shape in A, which is not perfectly symmetrical, is a better representation.
Step 5: Final Answer:
The path traced by the tip of the hour hand is correctly represented by the curve in option (A).