Question

На малому поршні розташований вантаж масою 10 кг, на великому — масою 160 кг. Діаметр ма- лого поршня гідравлічної машини дорівнює 4 см. Яким є діаметр великого поршня, якщо поршні перебувають у рівновазі та їхніми масами можна знехтувати?

Answer 1

Дано  m1=10 кг     m2=160 кг          d1=4 см    d2- ?

по закону Паскаля   P1=P2

F1/S1=F2/S2
S2/S1=F2/F1=m2*g/m1*g=16
так как S=π*d²/4
d2²/d1²=16
d2/d1=4
d2=4*d1=16 см

Ответ d2=16 см

Answer 2

Calculation of the Diameter of the Large Piston

To calculate the diameter of the large piston, we can use the principle of equilibrium. In a hydraulic system, the pressure exerted on the small piston is equal to the pressure exerted on the large piston. The pressure is given by the formula:

Pressure = Force / Area

Since the pistons are in equilibrium, the force exerted on the small piston is equal to the force exerted on the large piston. We can express this relationship as:

Force_small = Force_large

The force exerted on each piston can be calculated using the formula:

Force = Mass * Acceleration

Given that the mass of the small piston is 10 kg and the mass of the large piston is 160 kg, and assuming that the acceleration is due to gravity (9.8 m/s^2), we can calculate the forces:

Force_small = 10 kg * 9.8 m/s^2 = 98 N Force_large = 160 kg * 9.8 m/s^2 = 1568 N

Since the pressure is the same on both pistons, we can equate the forces divided by the areas:

Pressure_small = Pressure_large

The area of the small piston can be calculated using the formula:

Area = (pi * diameter^2) / 4

Given that the diameter of the small piston is 4 cm, we can calculate the area:

Area_small = (pi * (4 cm)^2) / 4 = 3.14 cm^2

Now, we can rearrange the equation to solve for the diameter of the large piston:

Area_large = (Pressure_small * Area_small) / Pressure_large

Substituting the known values:

Area_large = (Pressure_small * 3.14 cm^2) / Pressure_large

Since the pressure is the same on both pistons, we can simplify the equation:

Area_large = 3.14 cm^2

Finally, we can solve for the diameter of the large piston using the formula for the area:

Area_large = (pi * diameter_large^2) / 4

Rearranging the equation:

diameter_large^2 = (4 * Area_large) / pi

Substituting the known value for the area:

diameter_large^2 = (4 * 3.14 cm^2) / pi

Simplifying the equation:

diameter_large^2 = 4 cm^2

Taking the square root of both sides:

diameter_large = 2 cm

Therefore, the diameter of the large piston is 2 cm.