Chapter 37, Problem 1Q

A rod is to move at constant speed v along the x axis of reference frame S, with the rod’s length parallel to that axis. An observer in frame S is to measure the length L of the rod. Which of the curves in Fig. 37-15 best gives length L (vertical axis of the graph) versus speed parameter β?

Figure 37-15 Questions 1 and 3.

To determine

To find:

Which of the curves in Fig. 37-15 best represents length L (vertical axis of the graph) versus speed parameter β?

Answer to Problem 1Q

Solution:

The curve ‘c’ can best represent the length L and speed parameter $\beta .$

Explanation of Solution

1) Concept:

Here, we need to use the concept of length contraction as the length of rod is being measured from the inertial frame of reference while the rod is moving.

2) Formula:

$L=L_0 \sqrt{1-{\beta }^2 }$

3) Given:

i. A rod is moving at a constant speed v parallel to the x-axis is a reference frame S.

ii. The rod’s length is parallel to the x-axis.

iii. An observer in the reference frame S is measuring the length of the rod.

iv. Fig. 37-15 of the textbook with 5 curves of L versus β.

4) Calculation:

We have the equation for length L as

$L=L_0 \sqrt{1-{\beta }^2 }$

Here, $L_0$ is the length measured when the rod is not moving, that is, at time $t=0$.

Expanding the equation for L, we see that L has to be approximately inversely proportional to the β, and decreasing as β increases.

Curve a shows L increases linearly with β.

Curve b shows L increases with β.

Curve c shows that L decreases with β and is approximately inversely proportional to the β.

Curve d shows L decreases linearly with β.

Curve e shows L decreases with β and shows an exponential decrease with β.

So, among the given curves, the curve ‘c’ can best represent the function $\sqrt{1-{\beta }^2 }$.

Conclusion:

Using the equation of length contraction, we can identify the curve of best fit.