One piston of a hydraulic lift holds 1.40*10^3 kg. The other holds an ice block (density= 917 kg/m^3) that is 0.076 m thick. Find the first piston's area.
(Cross sectional area of the ice block = cross sectional area of the piston).
To find the area of the first piston, we can use the formula:
Area = Force / (Pressure)
We are given the force exerted by the piston, which is the weight of the object it is supporting. In this case, we are given the mass of the object held by the piston (1.40 * 10^3 kg). We can calculate the weight using the equation:
Weight = Mass * Acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2. Thus, the weight is:
Weight = (1.40 * 10^3 kg) * (9.8 m/s^2)
Next, we need to find the pressure exerted by the weight of the object on the piston. The pressure is equal to the weight divided by the area of the piston in contact with the object. We are told that the cross-sectional area of the ice block is the same as the piston. Therefore, the pressure can be expressed as:
Pressure = Weight / Area
We can rearrange this equation to solve for the area:
Area = Weight / Pressure
Substituting the weight and pressure values, we get:
Area = ((1.40 * 10^3 kg) * (9.8 m/s^2)) / Pressure
Now, let's find the pressure. The pressure exerted by the ice block can be calculated using the density of the ice and the thickness of the block. The pressure is given by the equation:
Pressure = Density * g * h
where density (ρ) is 917 kg/m^3 (density of the ice block), g is the acceleration due to gravity (9.8 m/s^2), and h is the thickness of the ice block, which is 0.076 m in this case.
Pressure = (917 kg/m^3) * (9.8 m/s^2) * (0.076 m)
Now, we can substitute the pressure value back into the equation to find the area:
Area = ((1.40 * 10^3 kg) * (9.8 m/s^2)) / ((917 kg/m^3) * (9.8 m/s^2) * (0.076 m))
Simplifying the equation, we find:
Area = (1.40 * 10^3 kg) / ((917 kg/m^3) * (0.076 m))
Now, calculate the area using the given values:
Area = 1.40 * 10^3 kg / (917 kg/m^3 * 0.076 m)
Area = 1.40 * 10^3 kg / 69.932 kg/m^2
Area ≈ 20.019 m^2
Therefore, the area of the first piston is approximately 20.019 m^2.
To find the area of the first piston, we can use the principles of fluid mechanics. In a hydraulic lift, the force applied to one piston is transmitted to the other piston through an incompressible fluid (usually oil). According to Pascal's law, the pressure is the same throughout the fluid in an enclosed system.
First, let's find the pressure acting on the piston that holds the ice block. We know the density of the ice block is 917 kg/m^3, and that the thickness of the ice block is 0.076 m. The weight of the ice block can be calculated as follows:
Weight = density * volume * acceleration due to gravity
= 917 kg/m^3 * (Area * thickness) * 9.8 m/s^2
The weight of the ice block is equal to the force acting on the second piston, which is also transmitted to the first piston. So, we can write:
Force = Mass * Acceleration
= 1.40 * 10^3 kg * 9.8 m/s^2
= 1.372 * 10^4 N
Since the pressure is the same throughout the system, we can equate the force acting on the second piston to the pressure times the area of the first piston:
Pressure * Area of piston 1 = Force
Therefore, we can solve for the area of piston 1:
Area of piston 1 = Force / Pressure
Substituting the values we have:
Area of piston 1 = 1.372 * 10^4 N / Pressure
Now, we can calculate the pressure acting on the piston that holds the ice block by using the weight of the ice block and the area of the ice block:
Pressure = (Weight of ice block) / (Area of ice block)
= (917 kg/m^3 * (Area of piston 1 * thickness) * 9.8 m/s^2) / (Area of piston 1)
Finally, we can substitute this value back into the equation for the Area of piston 1 to get the final answer.