Problem 34 - Physics

Question

A dentist's chair with a patient in it weighs \(2100 \mathrm{~N}\). The output plunger of a hydraulic system begins to lift the chair when the dentist's foot applies a force of \(55 \mathrm{~N}\) 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?

Step-by-Step Solution

Answer

The radius ratio \( \frac{r_2}{r_1} \) is approximately 6.18.

Step 1: Understand the hydraulic lift principle

A hydraulic system follows Pascal's principle, which means that the pressure applied to the fluid is transmitted undiminished throughout the fluid. The force applied to the input piston generates an equal pressure in the fluid, resulting in the lifting of the output plunger. The formula for pressure is given by: \( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \). Hence, the pressures on the input and output sides are equal: \( \frac{F_1}{A_1} = \frac{F_2}{A_2} \).

Step 2: Express force relation in terms of areas

From the equality of pressures, the forces and areas are related by: \( \frac{55}{A_1} = \frac{2100}{A_2} \). Thus, \( 55A_2 = 2100A_1 \).

Step 3: Relate area to radius

The area for the piston and plunger can be expressed in terms of their radii: \( A = \pi r^2 \). Therefore, \( A_1 = \pi r_1^2 \) and \( A_2 = \pi r_2^2 \).

Step 4: Substitute area relationship into the force equation

Substituting the expressions for areas into the force equation from Step 2: \( 55\pi r_2^2 = 2100\pi r_1^2 \). This simplifies to \( 55r_2^2 = 2100r_1^2 \).

Step 5: Solve for the ratio of radii

Rearrange the equation \( 55r_2^2 = 2100r_1^2 \) to find \( \frac{r_2^2}{r_1^2} = \frac{2100}{55} \), which simplifies to \( \frac{r_2}{r_1} = \sqrt{\frac{2100}{55}} \). Compute the square root to find \( \frac{r_2}{r_1} \approx 6.18 \).

Key Concepts

Pascal's Principle

When you step on the brake pedal of a car, ever wondered how such small force helps stop something as massive as a vehicle? This magic is thanks to Pascal's Principle, named after the French scientist Blaise Pascal. Simply put, Pascal's Principle states that when pressure is applied to a confined fluid, it is transmitted equally in all directions. This forms the backbone of how hydraulic systems work. In our example of the dentist's chair, when a force is applied to the input piston, it transmits the pressure through the hydraulic fluid to the output plunger. The equation that describes this is:

• Pressure = \( \frac{Force}{Area} \)

This means the pressure you apply at one point is felt across the whole system. So, the tiny force you apply is multiplied, allowing you to lift heavy weights like a dentist chair effortlessly. This concept is utilized in many areas, all thanks to the consistency of pressure across the system.

Pressure in Fluids

Imagine trying to squeeze a balloon; you will notice the air inside exerts pressure back on all sides. That's similar to how fluids behave in hydraulic systems. Pressure in a fluid is the force per unit area applied in a direction perpendicular to the surface of an object. In our dentist chair problem, when the dentist exerts a force of 55 N on the foot pedal, this pressure is uniformly distributed through the fluid to every part of the system. This principle is crucial because whether you're lifting a heavy chair or using a car jack, the fluid you are using has to distribute pressure evenly. In mathematical terms:

• Pressure = \( \frac{Force}{Area} \)

In this case, the 55 N applied creates a pressure that is exactly replicated at the larger plunger, letting it lift heavier objects without needing a direct application of large force at that point. This helps us understand how hydraulics can turn small force inputs into large outputs effortlessly.

Force and Area Relationship

If you've ever tried pressing a thumbtack with your thumb instead of your palm, you understand how area matters. With hydraulic systems, the force and area relationship guides how these systems amplify force. The basic idea is that force is distributed across the area, as described by the equation:

• Pressure = \( \frac{Force}{Area} \)
• If the area increases, so can the force, assuming pressure remains constant.

In the dental chair scenario, we have an input piston with a smaller area and an output plunger with a larger area. By making the plunger's area larger relative to the piston's, the same pressure leads to a larger force.Expressing area in terms of radius helps visualize this:

• Area = \( \pi r^2 \)

Thus, differences in radii directly influence the area's size and consequently, the force. Calculating this ratio \( \frac{r_2}{r_1} \), where \( r_2 \) is the plunger radius and \( r_1 \) is the piston's, shows how this balance results in increased force, making it possible to lift the patient smoothly onto the dental chair.