Question

9.1 * be sure you understand why a pendulum in equilibrium hanging in a car that is accelerating forward tilts backward, and then consider the following: a helium balloon is anchored by a massless string to the floor of a car that is accelerating forward with acceleration a. explain clearly why the balloon tends to tilt forward and find its angle of tilt in equilibrium. [hint: helium balloons float because of the buoyant archimedean force, which results from a pressure gradient in the air. what is the relation between the directions of the gravitational field and the buoyant force?]

Answer 1

The angle of tilt of the balloon in equilibrium is 45 degrees.

In equilibrium, the sum of all forces acting on the balloon must be zero. Therefore, we can write:

Buoyant force - Tension in the string - Gravitational force = 0

Substituting the expressions for each force, we get:

ρ_air * g * V - T - ρ_helium * g * V = 0

where:

ρ_air is the density of air

g is the acceleration due to gravity

V is the volume of the balloon

T is the tension in the string

ρ_helium is the density of helium

Solving for T, we get:

T = ρ_air * g * V - ρ_helium * g * V

The buoyant force is always directed upwards, while the gravitational force is always directed downwards. Therefore, the angle of tilt of the balloon is determined by the balance between these two forces.

To find the angle of tilt, we can use the following trigonometric relationship:

tan(θ) = (Buoyant force - Gravitational force) / Tension in the string

Substituting the expressions for each force, we get:

tan(θ) = (ρ_air * g * V - ρ_helium * g * V) / (ρ_air * g * V - ρ_helium * g * V)

Simplifying, we get: tan(θ) = 1 .