Final answer:
To determine the initial volume flow rate of oil through a hole, the velocity was found using Torricelli's Law and the flow rate was calculated by multiplying this velocity by the hole's cross-sectional area. The result was 1.91 liters per second.
Step-by-step explanation:
To calculate the initial volume flow rate of oil through a hole in a container, we need to apply the principles of fluid dynamics. More specifically, the Bernoulli's equation for incompressible fluids and the Hagen-Poiseuille law for viscous flow could be used here.
However, the arrangement suggests that we may treat this as a straightforward application of Torricelli's Law (akin to fluid exiting through an orifice), which doesn't account for viscosity directly. Since viscosity isn't ignored in general fluid dynamics but is for this specific case (a large tank and a small hole serve as a simplification), we proceed as follows:
- First, calculate the velocity (v) of the oil as it leaves the hole using Torricelli's Law:
v = √(2gh), where g is the acceleration due to gravity (9.81 m/s²) and h is the depth of the oil (2.86 m). - Second, once we have the velocity, we can compute the flow rate (Q) using the equation Q = Av, where A is the cross-sectional area of the hole. The area A can be found with the formula A = πr², where r is the radius of the hole (0.009 m).
- Multiply the velocity by the area to get the volume flow rate in m³/s, then convert it to liters per second by multiplying by 1000 (since 1 m³ = 1000 L).
Let's perform these calculations:
- Calculate the velocity (v):
v = √(2 * 9.81 m/s² * 2.86 m)
v ≈ 7.53 m/s - Calculate the cross-sectional area (A):
A = π * (0.009 m)²
A ≈ 2.54 * 10^-4 m² - Calculate the volume flow rate (Q):
Q = 7.53 m/s * 2.54 * 10^-4 m²
Q ≈ 1.91 * 10^-3 m³/s
Q ≈ 1.91 L/s
Therefore, the initial volume flow rate of oil through the hole is approximately 1.91 L/s.