A diagram shows two cylinders containing two pistonsA and B. The two cylinders are connected with a tube containing a fluid. The system is similar to the hydraulic braking system of a car. A man pushes down on piston A while piston B has a 10kg mass sitting on it. The surface area of piston A is 50 times larger than the surface area of piston B.
Pressure is transmitted through the fluid with no losses.
Show by calculation why the man is unable to lift 10kg of mass.
force = pressure * area
weight of 10 kg = 9.81*10 = 98.1 N
weight of man = maybe 100kg*9.81 = 981 N
981 /50A = pressure
force up on other end = pressure * A
= 981/50
= 19.62 Newtons
well he needs about 100 N and his whole weight only produces about 20 N even though he is a 220 pound man (about 100 kg)
To understand why the man is unable to lift the 10kg mass, we need to consider the principle of Pascal's law, which states that pressure applied to a fluid in a closed system is transmitted equally to all parts of the fluid and to the walls of the container.
Let's denote the surface area of piston A as A_A, and the surface area of piston B as A_B. Given that A_A is 50 times larger than A_B, we can express this relationship as:
A_A = 50A_B
According to Pascal's law, the pressure applied to the fluid by piston A is transmitted equally to piston B. Therefore, the pressure exerted on piston B, P_B, will be the same as the pressure applied on piston A, P_A.
Now, let's consider the forces acting on each piston. The force exerted on piston A, F_A, can be calculated using the formula:
F_A = P_A * A_A
Similarly, the force exerted on piston B, F_B, can be calculated as:
F_B = P_B * A_B
Since P_A = P_B, we can simplify these equations to:
F_A = P * A_A
F_B = P * A_B
where P is the pressure applied by the man.
Now, let's assume the man applies a force F_man on piston A, so we have:
F_man = F_A
Since the force on piston B is due to the 10kg mass, we can calculate it using the formula for gravitational force:
F_gravity = m * g
where m is the mass (10kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
However, we can also express F_gravity using the force on piston B:
F_gravity = F_B
Therefore, we have:
F_B = m * g
Now, let's substitute the expressions for F_A and F_B into this equation:
P * A_A = m * g
Now, let's substitute the relationship between A_A and A_B:
P * 50A_B = m * g
Dividing both sides of the equation by A_B, we get:
50P = m * g
Now, let's substitute the values given in the question:
50P = 10kg * 9.8 m/s^2
Simplifying the equation, we find:
P = 10kg * 9.8 m/s^2 / 50
P ≈ 1.96 N/m^2
Therefore, the required pressure to lift the 10kg mass would be approximately 1.96 N/m^2. However, the pressure the man can generate is limited by his physical strength, and it will most likely not be sufficient to produce the necessary pressure to lift the 10kg mass. Hence, the man is unable to lift the 10kg mass in this hydraulic system.