A glass bottle of soda is sealed with a screw cap. The absolute pressure of the carbon dioxide inside the bottle is 1.80 x10^5pa. Assuming that the top and bottom surfaces of the cap each have an area of .0004M^2, obtain the magnitude of the force that the screw thread exerts on the cap in order to keep it on the bottle. The air pressure outside the bottle is one atmosphere.
(Area)*(Poutside - Pinside) = Force
Pinside = 1.80*10^5 N/m^2
Poutside = 1 atm = 1.013*10^5 N/m^2
Area = 4*10^-4 m^2
Solve for Force, F
F = (4.0^-4m^2)(1.013*10^5 N/m^2 - 1.80*10^5 N/m^2)
F = (0.0004)(-78799)
F = -31.5196
N
To obtain the magnitude of the force that the screw thread exerts on the cap in order to keep it on the bottle, we can use the formula:
Force = (Area) x (Poutside - Pinside)
Where:
Area = 0.0004 m^2 (given)
Pinside = 1.80 x 10^5 Pa (given)
Poutside = 1 atm = 1.013 x 10^5 Pa (given)
Substituting the values into the formula:
Force = (0.0004 m^2) x (1.013 x 10^5 Pa - 1.80 x 10^5 Pa)
Calculating the difference in pressure:
Force = (0.0004 m^2) x (-78799 Pa)
Finally, solving for the force:
Force = -31.5196 N
Therefore, the magnitude of the force that the screw thread exerts on the cap in order to keep it on the bottle is approximately 31.52 N. Note that the negative sign indicates that the force is directed inward, towards the inside of the bottle.