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Chapter 15: Problem 58

A spherical rubber balloon with mass \(0.85 \mathrm{g}\) and diameter \(30 \mathrm{cm}\) is filled with helium (density \(0.18 \mathrm{kg} / \mathrm{m}^{3}\) ). How many 1.0 -g paper clips can you hang from the balloon before it loses buoyancy?

Short Answer

Expert verified
The solution to the problem must be calculated using the relevant values for mass, volume, density, and the weight of one paper clip. To indicate how many paper clips you can hang from the balloon before it loses buoyancy, you must use the appropriate formulae and then solve for the unknown. The final answer will be the calculated number of paper clips.

Step by step solution

01

Calculate Volume of the Balloon

The volume of a sphere can be calculated using the formula \( V = \frac{4}{3}\pi r^{3} \) where r is the radius of the sphere. The problem specifies the diameter of the balloon as 30 cm, which means the radius is 15 cm or 0.15 m. Thus the volume will be \( V = \frac{4}{3}\pi (0.15)^{3} \) m^{3}.
02

Calculate Buoyant Force on the Balloon

The buoyant force can be calculated using the formula \( F_{b} = \rho_{f}gV \), where \( \rho_{f} \) is the density of the fluid (air in this case), g is the acceleration due to gravity and V is the volume of the object. We are not given the density of air, however it is typically about 1.29 kg/m^{3}. The acceleration due to gravity is approximately 9.8 m/s^{2}. Thus the buoyant force is \( F_{b} = 1.29 * 9.8 * V \) N.
03

Calculate the Weight that the Balloon can Lift

We know that the lift of the balloon must equal to the buoyant force, which equals the weight of the air displaced minus the weight of the balloon and the helium inside it. Weight can be calculated as mass times gravity. So the mass lifted by the balloon (m) refers to the mass of helium (m_{H}) plus the mass of the balloon (m_{B}) plus the mass the paper clip adds (m_{P}). That is, \( m_{B} + m_{H} + m_{P} = F_{b}/g \). Substituting the known values: \( m_{H} = \rho_{H}*V; m_{P} = (F_{b}/g) - m_{B} - m_{H} \), where \( \rho_{H} \) is the density of helium, which is given as 0.18 kg/m^{3}.
04

Calculate the Number of Paper Clips

Since we know each paper clip weighs 1.0 g (or 0.001 kg), the number of paper clips n is simply the total mass the paper clip adds \( m_{P} \) divided by the weight of one paper clip. \( n = m_{P}/0.001 \).

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