Question

A person is about to take off in an air balloon, full of helium, can be considered as a sphere with a radius of 7.5 m. The basket and balloon material have a combined mass of 314 kg. The balloon is kept from moving up by a certain amount of ballast, in the form of sandbags, which provides its own weight. The person has a mass of 125 kg. Helium has a density of 0.18 kg/m3, while air has a density of 1.29 kg/m3.

A) draw a diagram showing the forces that are acting on the hot air balloon.

B) what is the volume of the air that the balloon displaces? why can you say this?

C) what is the mass of the helium in the balloon?

D) what is the combined weight of the person, the basket, balloon, and the helium?

E) what is the buoyant force of the air acting on the balloon?

F) how many kilograms of ballast are required to hold the balloon motinless, the person can get aboard?

G) if the ballast is cut loose, what will be the upward acceleration of the balloon?

Answer 1

A) ballon has a downward force of its own weight from the mass of helium and basket = (m_h + mb)*g

= (314 + m_h)*g

It also has the persons weight = 125*g

It also has a buyont force = weight of the air displaced upwards on the balloon.

And weight of sand bags acting downward to restrict its motion.

If its touching the groung, then we have normal reaction

B)Volume displaced = Volume of the helium balloon.  

Imagine a closed space contains V volume of gas A. To fill another gas into the same shape, We need to remove V volume of gas A and fill it with gas B.

Similarly in this vast universe, air occupies certain volume. We are mixing some helium in the air. So the same volume of air will be taken and moved some where wlse. So the same volume of helium is displaced.

So answer for B = 4/3*pi*7.5^3 = 1767.15 m^3

C)Mass = volume 8 density = 1767.15*0.18 = 318.08kg

D) Total weight = 125+314+318.08 = 757.08 kg

E) Buoyant force = weigth of the air displaced = volume displaced * density of air = 1767.15*1.29 = 2279.62 Kg

F)So to hold the ballon, weight of the ballast should be 2279.62 - 757.08 = 1522.54 Kg

G) If it is cut loose, acceleration will be 1522.54*9.8/757.08 = 19.70m/s^2