Piston 1 has a diameter of 0.643 cm.
Piston 2 has a diameter of 4.59 cm. In the absence of friction, determine the force required on piston 1 necessary to support an object, m, with weight 1151 N. (Neglect the height difference between the bottom of the two pistons, and assume that the pistons are massless).
What is the magnitude of the force, F, required at the end of the handle?
Well, first of all, let me just say that these pistons sure know how to make an entrance! It's like they're auditioning for a talent show or something!
Now, to answer your question, let's have some fun with physics!
We can start by calculating the area of piston 1 using its diameter, which is 0.643 cm. Now, I know you didn't specifically ask for the area, but trust me, it's going to come in handy later. So, the area of piston 1 is given by:
A1 = π * (diameter/2)^2
Just plug in the values and calculate away!
Next, we can find the area of piston 2, which has a diameter of 4.59 cm. Again, using the same formula, we get:
A2 = π * (diameter/2)^2
Now, let's put some weight on these poor pistons! The object, m, has a weight of 1151 N. Since the pistons are massless, we can assume that the force required to balance the weight on piston 1 is equal to the weight itself. So:
Force1 = 1151 N
Now, let's find the force required at the end of the handle, F. We know that the pressure on each piston is the force applied divided by the area of the piston. So, for piston 1:
Pressure1 = Force1 / A1
Similarly, for piston 2:
Pressure2 = Force2 / A2
Since the pressure is the same at both pistons (neglecting friction), we can equate the two pressures:
Pressure1 = Pressure2
And rearranging the equation, we find:
Force2 = Pressure2 * A2
Now, remember, we want to find the force at the end of the handle, F. And F is the force required to support the weight on piston 2. So, we can write:
F = Force2
Now just substitute the values we have into the equation and calculate the force!
Voilà! There you have it. I hope this humorous approach to physics made it a little more enjoyable for you!
To determine the magnitude of the force required at the end of the handle, we can use the principle of Pascal's Law, which states that the pressure applied to a fluid in a closed system is transmitted equally in all directions.
Given:
- Piston 1 diameter (d1) = 0.643 cm = 0.00643 m
- Piston 2 diameter (d2) = 4.59 cm = 0.0459 m
- Weight of the object (m) = 1151 N (since weight = mass * gravity, where gravity is approximately 9.8 m/s^2)
Step 1: Calculate the area of the two pistons.
The area of a piston can be calculated using the formula for the area of a circle: A = π * r^2, where r is the radius of the piston.
- Piston 1 area (A1) = π * (d1/2)^2
= π * (0.00643/2)^2
- Piston 2 area (A2) = π * (d2/2)^2
= π * (0.0459/2)^2
Step 2: Calculate the pressure.
Since Pascal's Law states that the pressure is transmitted equally, the pressure applied by piston 2 would be the same as the pressure required to support the object.
- Pressure (P) = weight / Piston 2 area
= 1151 / A2
Step 3: Calculate the force required on piston 1.
Using the formula for force (F = P * A), we can calculate the force required on piston 1.
- Force (F) = P * A1
= (1151 / A2) * A1
Calculating the values and simplifying the equation will give you the magnitude of the force required at the end of the handle.
To determine the magnitude of the force required at the end of the handle, we need to consider the principles of Pascal's law and the relationship between pressure and force in a hydraulic system.
Pascal's law states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. In a hydraulic system, we have two pistons of different sizes connected by an incompressible fluid.
First, let's calculate the area of each piston using their respective diameters:
Area1 = π * (radius1)^2
= π * (0.643 cm / 2)^2
Area2 = π * (radius2)^2
= π * (4.59 cm / 2)^2
Next, we need to find the pressure exerted by the weight of the object on piston 2. Since the pistons are in equilibrium, the pressure on piston 1 needs to be equal to the pressure on piston 2.
The pressure exerted by the weight of the object (P2) can be calculated using the equation:
P = F / A
where P is the pressure, F is the force, and A is the area.
P2 = (Weight of object) / Area2
Substituting the values, we get:
P2 = 1151 N / Area2
Once we have the pressure on piston 2, we can use Pascal's law to determine the force required on piston 1. Since the pressure is transmitted undiminished, the pressure on piston 1 (P1) is also equal to P2.
P1 = P2
Finally, we can calculate the force required on piston 1 (F1) using the equation:
F1 = P1 * Area1
= P2 * Area1
Substituting the values, we get:
F1 = P2 * Area1
By entering the given values and following these steps, you will be able to determine the magnitude of the force required at the end of the handle.
ratio of areas = D1^2/D2^2 = .643^2/4.59^2 = .0196
.0196*1151 = 22.6 N