Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 28a

Chapter 3, Problem 28a

The minute hand on a watch is 2.0 cm in length. What is the displacement vector of the tip of the minute hand in each case? Use a coordinate system in which the y-axis points toward the 12 on the watch face. From 8:00 to 8:20 a.m.

Verified step by step guidance

Step 1: Understand the problem. The minute hand rotates around the center of the watch face, and we need to calculate the displacement vector of its tip as it moves from 8:00 to 8:20 a.m. The displacement vector is the straight-line distance and direction from the initial position to the final position of the tip of the minute hand.

Step 2: Define the coordinate system. The center of the watch face is the origin (0, 0). The y-axis points toward the 12 on the watch face, and the x-axis points toward the 3. The minute hand is 2.0 cm long, so its tip lies on the circumference of a circle with radius 2.0 cm.

Step 3: Determine the initial position of the tip of the minute hand at 8:00. At 8:00, the minute hand points to the 12 on the watch face, which corresponds to the positive y-axis. The coordinates of the tip are (0, 2.0).

Step 4: Determine the final position of the tip of the minute hand at 8:20. At 8:20, the minute hand points to the 4 on the watch face. The angle between the 12 and the 4 is 120 degrees clockwise. Convert this angle to radians: $ \theta = 120 \times \frac{\pi}{180} $. Use trigonometry to find the coordinates of the tip: $ x = r \cdot \cos(\theta) $ and $ y = r \cdot \sin(\theta) $, where $ r = 2.0 \, \text{cm} $.

Step 5: Calculate the displacement vector. Subtract the initial position vector from the final position vector: $ \vec{d} = (x_{\text{final}} - x_{\text{initial}}, y_{\text{final}} - y_{\text{initial}}) $. This gives the components of the displacement vector.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Displacement Vector

A displacement vector represents the change in position of an object from its initial to its final location. It is defined by both magnitude and direction. In this context, the displacement vector of the minute hand will indicate how far and in which direction the tip of the hand moves as time progresses.

Circular Motion

The minute hand of a watch moves in a circular path around the center of the watch face. This motion can be described using angular displacement, which measures the angle through which the hand moves. Understanding circular motion is essential for calculating the position of the minute hand at different times.

Coordinate System

A coordinate system provides a framework for defining the position of points in space. In this problem, a Cartesian coordinate system is used where the y-axis points toward the 12 on the watch face. This system helps in determining the coordinates of the tip of the minute hand at specific times, facilitating the calculation of the displacement vector.

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