A dentist's chair with a patient in it weighs 2180 N. The output plunger of a hydraulic system begins to lift the chair, when the dentist's foot applies a force of 73.6 N to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston?
R1^2/R2^2 = 2180/73.6
so
sqrt (2180/73.6)
Well, I may not have a dental degree, but I can certainly help with this question! To solve for the ratio of the radii, we can use Pascal's law, which states that the pressure applied to an enclosed fluid will be transmitted undiminished to all portions of the fluid and to the walls of its container. Here, we have a hydraulic system, so let's dive in!
The pressure applied to the input piston is equal to the pressure applied to the output plunger. The formula for pressure is P = F/A, where P is pressure, F is force, and A is area. Since the area is directly proportional to the square of the radius, we can say that the ratio of the radii will be equal to the square root of the ratio of forces.
Let's do the math: the force applied to the input piston is 73.6 N, while the force exerted on the output plunger is the weight of the dentist's chair, which is given as 2180 N.
So, the ratio of the radii will be the square root of the ratio of forces: √(2180 N / 73.6 N).
Calculating this, we find that the ratio of the radii is approximately 8.82.
Therefore, the ratio of the radius of the plunger to the radius of the piston is about 8.82. You could say the plunger has quite an "inflated" ego compared to the humble piston!
To find the ratio of the radius of the plunger to the radius of the piston, we can use Pascal's law, which states that the pressure exerted on a fluid in a confined space is transmitted undiminished in all directions. In this case, the force exerted on the input piston is equal to the force exerted on the output plunger.
Let's denote the radius of the plunger as r_p and the radius of the piston as r_pis. The input force, F_in, is 73.6 N, and the output force, F_out, is the weight of the chair, which is 2180 N.
According to Pascal's law, the pressure exerted on the fluid is the same at both pistons, which we can express using the formula:
P_in = P_out
The pressure at the input piston can be calculated as:
P_in = F_in / A_in,
where A_in is the area of the input piston, given by A_in = π * r_in^2.
Similarly, the pressure at the output plunger can be calculated as:
P_out = F_out / A_out,
where A_out is the area of the output plunger, given by A_out = π * r_out^2.
Since P_in = P_out, we have:
F_in / A_in = F_out / A_out.
Substituting the expressions for A_in and A_out, we get:
F_in / (π * r_in^2) = F_out / (π * r_out^2).
Simplifying the equation, we have:
F_in / r_in^2 = F_out / r_out^2.
Since F_in = 73.6 N and F_out = 2180 N, we can substitute these values to obtain:
73.6 / r_in^2 = 2180 / r_out^2.
To find the ratio of the radii, we can divide both sides of the equation:
r_in^2 / r_out^2 = 2180 / 73.6.
Taking the square root of both sides gives:
r_in / r_out = √(2180 / 73.6).
Calculating this expression, we find:
r_in / r_out ≈ 3.928.
Therefore, the ratio of the radius of the plunger to the radius of the piston is approximately 3.928.
To find the ratio of the radius of the plunger to the radius of the piston, we can use the principle of Pascal's law, which 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 this case, the force applied to the input piston (73.6 N) results in an increase in pressure in the hydraulic fluid, which is transmitted to the output plunger lifting the chair.
We can start by calculating the pressure exerted by the input piston. Pressure (P) is defined as the force applied (F) divided by the area over which it is applied (A). Mathematically, we can express it as:
P(input) = F(input) / A(input)
Similarly, we can calculate the pressure exerted by the output plunger. Since the pressure is transmitted undiminished, it is the same throughout the fluid, including the output plunger. Therefore, we can express it as:
P(output) = F(output) / A(output)
Since the only force acting on the output plunger is the weight of the chair (2180 N), we can write:
P(output) = 2180 N / A(output)
According to Pascal's law, P(input) = P(output) in a closed hydraulic system. Therefore, we have:
F(input) / A(input) = 2180 N / A(output)
Rearranging the equation, we get:
A(output) / A(input) = F(input) / 2180 N
Now, let's introduce the relationship between the radii of the pistons to their areas. The area of a piston is given by the formula A = πr^2. Therefore, we can say:
A(output) = πr(output)^2
A(input) = πr(input)^2
Substituting these expressions, we get:
πr(output)^2 / πr(input)^2 = F(input) / 2180 N
Simplifying further:
(r(output)^2) / (r(input)^2) = F(input) / 2180 N
Taking the square root of both sides to isolate the ratio of the radii:
(r(output) / r(input)) = sqrt(F(input) / 2180 N)
Now we can substitute the values given in the question:
r(output) / r(input) = sqrt(73.6 N / 2180 N)
Calculating this expression:
r(output) / r(input) = sqrt(0.0338)
Finally, we have the ratio of the radius of the plunger to the radius of the piston:
r(output) / r(input) ≈ 0.183
Therefore, the ratio of the radius of the plunger to the radius of the piston is approximately 0.183.