Question
In the hydraulic pistons shown in the sketch, the small piston has a diameter of 2 cm; the large has a diameter of 6 cm. How much more force can the larger piston exert compared with the force applied to the smaller piston
Answers
GPT 3.5
To compare the force exerted by the two pistons, we can use the concept of pressure in a hydraulic system. The pressure in a hydraulic system is the same throughout, so we can set the pressure on the smaller piston equal to the pressure on the larger piston:
Pressure on smaller piston = Pressure on larger piston
The formula for pressure is:
Pressure = Force / Area
The area of the smaller piston is:
Area of smaller piston = π * (radius)^2
= π * (1 cm)^2
= π cm^2
The area of the larger piston is:
Area of larger piston = π * (radius)^2
= π * (3 cm)^2
= 9π cm^2
Let F1 be the force exerted by the smaller piston and F2 be the force exerted by the larger piston. The equation for pressure is:
F1 / (π cm^2) = F2 / (9π cm^2)
F1 = F2 / 9
Therefore, the larger piston can exert 9 times more force compared to the force applied to the smaller piston.
Pressure on smaller piston = Pressure on larger piston
The formula for pressure is:
Pressure = Force / Area
The area of the smaller piston is:
Area of smaller piston = π * (radius)^2
= π * (1 cm)^2
= π cm^2
The area of the larger piston is:
Area of larger piston = π * (radius)^2
= π * (3 cm)^2
= 9π cm^2
Let F1 be the force exerted by the smaller piston and F2 be the force exerted by the larger piston. The equation for pressure is:
F1 / (π cm^2) = F2 / (9π cm^2)
F1 = F2 / 9
Therefore, the larger piston can exert 9 times more force compared to the force applied to the smaller piston.