A pond has the shape of an ice cream cone with the tip slices off and has a depth of 5.00m. The atmospheric pressure above the pond is 1.01 *10^5 Pa. The circular top surface (radius = R2) and circular bottum (radis = R1) of the pond are both parrallel to the ground. The angle between the ground and the side of the pond is 60 degrees. The magnitude of the force acting on the top surface is the same as the magnitude of the force acting on the bottum surface. Calculate R1 and R2.
Help please, I am currently manipulating formulas, but this seems extremely complicated, can someone show me the process to which they got the answer and I will compare it with my way. Thanks.
You need to solve two simultaneous equations for R1 and R2.
The first is the geometrical relation:
R2 = R1 + 5/tan 60 = R1 + 2.89 m
The second is the physics relation:
pi*R2^2 * Po =
pi*R1^2 (Po + rho*g*depth)
Po = atmospheric pressure = 1.01*10^5 Pa
rho*g*depth = 0.49*10^5 Pa
R2^2/R1^2 = 1.50/1.01 = 1.49
R2/R1 = 1.22
Take it from there.
o wow, i didn't even think of the geometrical way, thanks.
To find the radii of the circular top and bottom surfaces of the pond, we can start by understanding the forces acting on these surfaces.
1. The force acting on the top surface:
The force acting on the top surface of the pond is due to the atmospheric pressure. This force can be calculated using the formula:
Force = Pressure × Area
Given that the atmospheric pressure is 1.01 × 10^5 Pa and the area of the circular top surface is πR2^2, we can express this force as:
Force (on top surface) = (1.01 × 10^5) × πR2^2 ------ Equation 1
2. The force acting on the bottom surface:
The force acting on the bottom surface is the weight of the water in the pond. Since the shape of the pond resembles an ice cream cone with the tip sliced off, we can imagine the bottom surface as the base of this cone.
The weight of the water is given by the formula:
Weight = Mass × Gravity
The mass of the water can be calculated using the formula:
Mass = Volume × Density
The volume of the water in the pond can be expressed as the volume of a frustum of a cone with depth 5.00m. The formula to calculate the volume of a frustum of a cone is:
Volume = (⅓)πh(R1^2 + R2^2 + R1R2)
Given that the density of water is approximately 1000 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2, we can express the weight of the water as:
Weight (of water) = (⅓)πh(R1^2 + R2^2 + R1R2) × 1000 × 9.8 ------ Equation 2
Now, since we are told that the magnitude of the force acting on the top surface is the same as the magnitude of the force acting on the bottom surface, we can equate Equations 1 and 2:
(1.01 × 10^5) × πR2^2 = (⅓)πh(R1^2 + R2^2 + R1R2) × 1000 × 9.8
Simplifying this equation will allow us to solve for R1 and R2.