What force must be exerted on the master cylinder of a hydraulic lift to support wt of a 2000 kg car resting on the slave cylinder? master : 2cm diameter,slave: 24cm diameter

13.9

To find the force needed to support the weight of the car resting on the slave cylinder, we first need to understand the principle of Pascal's law for hydraulic systems.

Pascal's law states that when there is an applied pressure to an enclosed fluid, the pressure is transmitted equally in all directions. In other words, any change in pressure applied to a fluid in a confined space will be transmitted undiminished to all portions of the fluid and to the walls of its container.

In this case, the hydraulic system consists of a master cylinder and a slave cylinder. The master cylinder has a diameter of 2 cm and the slave cylinder has a diameter of 24 cm. We can assume that the system is filled with an incompressible fluid, such as oil.

To determine the force exerted on the master cylinder, we need to compare the areas of the master and slave cylinders and calculate the ratio between them.

The area of a cylinder can be calculated using the formula: A = π * r^2, where A is the area, and r is the radius.

Let's calculate the area of the master cylinder:
Radius of the master cylinder = diameter / 2 = 2 cm / 2 = 1 cm = 0.01 m
Area of the master cylinder = π * (0.01 m)^2 = 0.000314 m^2

Now, let's calculate the area of the slave cylinder:
Radius of the slave cylinder = diameter / 2 = 24 cm / 2 = 12 cm = 0.12 m
Area of the slave cylinder = π * (0.12 m)^2 = 0.045 m^2

Since the pressure is transmitted equally in all directions, the pressure exerted on the master cylinder is the same as the pressure exerted on the slave cylinder. Therefore, we can set up a ratio between the forces exerted on the master and slave cylinders:

Force on master cylinder / Area of master cylinder = Force on slave cylinder / Area of slave cylinder

We want to find the force on the master cylinder (which is what is exerted), and we know the area of the master cylinder and the area of the slave cylinder. Rearranging the equation, we get:

Force on master cylinder = (Force on slave cylinder / Area of slave cylinder) * Area of master cylinder

The force on the slave cylinder is the weight of the car. The weight can be calculated using the formula: weight = mass * gravitational acceleration.

Given:
Mass of the car = 2000 kg
Gravitational acceleration = 9.8 m/s^2

Weight of the car = 2000 kg * 9.8 m/s^2 = 19600 N

Now, let's substitute the values into the equation to find the force on the master cylinder:

Force on master cylinder = (19600 N / 0.045 m^2) * 0.000314 m^2
Force on master cylinder ≈ 6867.11 N

Therefore, a force of approximately 6867.11 Newtons must be exerted on the master cylinder of the hydraulic lift to support the weight of a 2000 kg car resting on the slave cylinder.

To calculate the force exerted on the master cylinder, we can use the principle of Pascal's law, which states that the pressure in a confined fluid is transmitted equally in all directions.

Step 1: Calculate the area of the master cylinder:
The formula for calculating the area of a circle is A = π * r^2, where A is the area and r is the radius. Since we are given the diameter, we need to divide it by 2 to get the radius.
Radius of the master cylinder (r_m) = 2cm / 2 = 1cm = 0.01m
Area of the master cylinder (A_m) = π * (0.01m)^2

Step 2: Calculate the area of the slave cylinder:
Radius of the slave cylinder (r_s) = 24cm / 2 = 12cm = 0.12m
Area of the slave cylinder (A_s) = π * (0.12m)^2

Step 3: Calculate the ratio of the areas:
A_ratio = A_m / A_s

Step 4: Determine the force on the master cylinder:
Since the pressure is transmitted equally in all directions, the force exerted on the master cylinder is equal to the force exerted on the slave cylinder multiplied by the area ratio.
Force exerted on the slave cylinder (F_s) = weight of the car = mass of the car * gravitational acceleration = 2000kg * 9.8m/s²
Force exerted on the master cylinder (F_m) = F_s * A_ratio

Following these steps, you can calculate the force exerted on the master cylinder.