for the hydrulic lift shown,what must be the radio of the diameter of the vessel at the car to the diameter of the vessel where the force F,is applied so that a 1.52x10^3 gr car can be lifted with a force F,of just 0.125kN?
To determine the ratio of the diameter of the vessel at the car (D_car) to the diameter of the vessel where the force (F) is applied (D_force), we can use the principle of Pascal's Law for hydraulic systems.
Pascal's Law states that when a pressure is applied to an enclosed fluid, the pressure is transmitted equally in all directions.
In this case, we can assume that the hydraulic lift operates under ideal conditions, meaning there are no energy losses due to friction or other factors.
Let's start by converting the given force from kilonewtons to newtons:
F = 0.125 kN = 0.125 × 1000 N = 125 N
Next, we need to convert the mass of the car from grams to kilograms:
mass = 1.52 × 10^3 grams = 1.52 × 10^3 kg (since 1 kg = 1000 grams)
Now, we can calculate the pressure applied by the force (F). Since the pressure is defined as the force divided by the area, we can determine the area using the formula:
pressure = force / area
125 N = pressure × area
We know that the pressure in the hydraulic system is transmitted equally, so the pressure at the car is the same as at the point where the force is applied.
Now, let's consider the area of the vessel at the car (A_car) and the area at the point where the force is applied (A_force). The area is given by the formula for the area of a circle:
Area = π × (radius)^2
Let's assume the radius of the vessel at the car is R_car, and the radius of the vessel where the force is applied is R_force.
Since we want to find the ratio of D_car (diameter of the vessel at the car) to D_force (diameter of the vessel where the force is applied), we can use the relationship between diameter and radius:
D_car = 2 × R_car
D_force = 2 × R_force
Substituting these values into the equation for pressure, we get:
125 N = pressure × (π × R_car^2)
125 N = pressure × (π × R_force^2)
Since the pressure is the same at both points, we have the equation:
π × R_car^2 = π × R_force^2
Now, we can simplify the equation further by canceling out the common factor (π):
R_car^2 = R_force^2
Taking the square root of both sides, we get:
R_car = R_force
Therefore, the ratio of the diameter of the vessel at the car to the diameter of the vessel where the force F is applied is 1:1. In other words, the diameters are equal.