Pascals Law

Step 1: Understanding the Concept:
According to the fundamental definition of fluid pressure, pressure is mathematically defined as the magnitude of the normal force applied per unit surface area.
When two connected or identical systems operate under the exact same fluid pressure, the required force becomes strictly directly proportional to the cross-sectional area of the specific piston.
Step 2: Key Formula or Approach:
The formula relating pressure , force , and area is universally given by .
Since the pressure is constant across both syringes, , which algebraically rearranges to , meaning the ratio of forces is equal to the ratio of areas: .
Step 3: Detailed Explanation:
The problem states that the pistons are in the strict ratio of .
In physics problems involving hydraulic lifts and interconnected syringes, a "ratio of pistons" typically refers directly to their effective cross-sectional areas (unless explicitly designated as radii or diameters).
Assuming the given ratio perfectly represents the area ratio, we have:
Because the pressure exerted is entirely identical for both systems ( ), we can substitute the area ratio directly into our derived force relationship:
Therefore, the necessary force applied on the first piston is one-fifth of the force strictly required on the second piston.
Step 4: Final Answer:
The ratio of the applied forces is .

The de Broglie wavelength of a particle is given by the equation: where is Planck's constant and is the momentum of the particle. The kinetic energy of a particle is related to its momentum as: where is the mass of the particle. Rearranging the equation for , we get: Since the kinetic energy is the same for both the proton and the electron, we can write the wavelengths for the proton and electron as: where and are the masses of the electron and proton, respectively. The ratio of the wavelengths is: Since (mass of proton) is approximately 1831 times greater than (mass of electron), we have: Thus, the relation between the wavelengths is: