Question

A rod at rest on Earth makes an angle of \(10^{\circ}\) with the \(x\) -axis. If the rod is moved along the \(x\) -axis, what happens to this angle, as viewed by an observer on the ground?

Short answer

Answer: The change in the angle is \(0^{\circ}\).

Step-by-step solution

Understand the initial situation

The rod is initially at rest. In this situation, it makes an angle of \(10^{\circ}\) with the x-axis. We will denote this initial angle by \(\theta_{initial}\), so \(\theta_{initial} = 10^{\circ}\).

Calculate the rod's movement along the x-axis

Let's assume the rod is moved along the x-axis by a distance \(d\). Due to the motion along the x-axis, the position of the rod changes, but its orientation remains the same. Therefore, the angle between the rod and the x-axis will not change.

Assess the rod's position from an observer's perspective

The observer on the ground will see the rod being moved along the x-axis. The rod's orientation does not change, so the angle between the rod and the x-axis remains the same from the observer's perspective, i.e., \(10^{\circ}\).

Determine the change in angle

For an observer on the ground, the angle between the rod and the x-axis remains the same, even after the movement along the x-axis, i.e., it remains at \(10^{\circ}\). Therefore, the change in angle, as viewed by an observer on the ground, can be calculated as: \(\Delta\theta = \theta_{final} - \theta_{initial}\), where \(\theta_{final} = 10^{\circ}\) and \(\theta_{initial} = 10^{\circ}\). Using these values, we get: \(\Delta\theta = 10^{\circ} - 10^{\circ} = 0^{\circ}\).

Conclusion

The change in the angle between the rod and the x-axis, as viewed by an observer on the ground, is \(0^{\circ}\). This implies that the angle does not change during the rod's movement along the x-axis.