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Chapter 3: Problem 76

Demonstrate with a vector diagram that a displacement is the same when measured in two different reference frames that are at rest with respect to each other.

Short Answer

Expert verified
Explain your answer. Answer: Yes, the displacement of an object remains the same when measured in two different reference frames that are at rest with respect to each other. This is because both frames have the same relative motion, and when we compare the displacement vectors in both reference frames, they have the same magnitude and direction.

Step by step solution

01

Define the displacement vector in the first reference frame

Let's consider an object that moves from point A to point B, and we have two reference frames R1 and R2, both at rest with respect to each other. We can represent the displacement of the object in reference frame R1 as a vector \(\vec{D1}\).
02

Define the displacement vector in the second reference frame

Similarly, we can represent the displacement of the object in reference frame R2 as a vector \(\vec{D2}\).
03

Create the vector diagram for R1

Now, let's create a vector diagram for reference frame R1. Draw a coordinate system with origin O1 and plot points A and B in this coordinate system, representing the initial and final positions of the object, respectively. Connect points A and B with an arrow, representing vector \(\vec{D1}\).
04

Create the vector diagram for R2

Create a vector diagram for reference frame R2. Draw a coordinate system with origin O2, and plot points A and B in this coordinate system, representing the initial and final positions of the object, respectively. Connect points A and B with an arrow, representing vector \(\vec{D2}\).
05

Compare the displacement vectors

Since both reference frames R1 and R2 are at rest relative to each other, the displacement vectors \(\vec{D1}\) and \(\vec{D2}\) should have the same magnitude and direction. Comparing the vector diagrams, we can see that the displacement vectors have the same length and are parallel. Hence, the displacement of the object is the same when measured in two different reference frames that are at rest with respect to each other.

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