Chapter 9, Problem 9.1P

To determine

The reason for the balloon to tilt forward and also the angel of the tilt in equilibrium.

Answer to Problem 9.1P

The angel of the tilt in equilibrium is $\theta ={\tan }^{-1}\left(\frac{A}{g}\right)$

Explanation of Solution

When the car is at the rest the balloon floats up in a direction opposite to the pull of the gravity $g$.when the car accelerates, the car defines a frame of references with some acceleration $a_0$ to the right.

From the equivalence principle, a frame accelerating to the right is equivalent to a non- accelerating frame with an artificial gravity $g_0$ to the left, where $g_0=a_0$. In this non-accelerating frame, there are actually two gravitates, the real gravity $g$ downward and the artificial gravity $g_0$ to the left.

Then the effective gravity is the vector sum of these two and directs as shown in the figure 1.

The balloon will float in a direction opposite to this $g_{\text{eff}}$ as shown in the figure 1. The balloon will move forward tilting towards the front of the car as shown in the figure 2.

 By the equivalence Principle, it will do the same thing in the actual, accelerating car towards the right.

Conclusion:

Write the expression from the figure 3,

  $\begin{array}{c}\tan \theta =\frac{-mA}{m\left(-g\right)}\\ \theta ={\tan }^{-1}\left(\frac{A}{g}\right)\end{array}$

Here, $m$ is the mass of the helium balloon, $A$ is the acceleration of the helium balloon, $g$ is the true gravity on the helium balloon and $g_{eff}$ is the effective gravity on the helium balloon.

The angel of the tilt in equilibrium is $\theta ={\tan }^{-1}\left(\frac{A}{g}\right)$