91Ó°ÊÓ

Hooke's Law is fundamental when dealing with springs and elastic materials. This law tells us how much force is needed to compress or stretch a spring by a certain amount. The formula for Hooke's Law is:

• \( F_s = kx \)

Here, \( F_s \) is the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the distance that the spring is compressed or stretched from its rest position.
For the exercise in question, the spring constant is given as \( 3.75 \times 10^4 \, \text{N/m} \). When you compress this spring by \( 5.00 \, \text{cm} \) or \( 0.050 \, \text{m} \), the force needed is \( 1875 \, \text{N} \). This is calculated by multiplying the spring constant by the compression distance. Understanding this helps us predict how springs will behave in different situations.

A hydraulic lever is a fantastic application of fluid mechanics principles. It allows us to amplify force using fluid pressure in a confined space. In this exercise, the setup involves input and output pistons connected by a fluid-filled chamber.

• The input piston is where force is applied.
• The output piston experiences a transmitted force.

The input and output pistons have different areas, which is a key feature of how a hydraulic lever works. In this scenario, the area of the output piston is \(18\) times that of the input piston. This means that any force applied on the input piston will be distributed over a much larger area on the output piston, effectively increasing the force by a factor of \(18\). This multiplication of force is what makes hydraulic systems so powerful and widely used in machines, like car brakes and excavators.

Pascal's Principle is a crucial concept for understanding the behavior of fluids and is the cornerstone of hydraulic systems. This principle states that any change in pressure applied to an enclosed fluid is transmitted equally throughout the fluid. In simpler terms:

• If you apply pressure to one part of a fluid, it gets transferred in all directions.

In a hydraulic lever, this principle allows force applied on the input piston to be transferred efficiently to the output piston. In our exercise, the pressure applied by the force \( F_i \) on the smaller input area \( A_i \) is transferred to the larger output area \( A_o = 18 A_i \).
Hence, when the input piston experiences a certain force, the output piston experiences an amplified force due to its larger area, allowing the spring to be compressed by the desired amount efficiently.

Gravitational force is the attractive force between any two masses. On Earth, this manifests as the weight of objects. The gravitational force acting on an object is calculated using the formula:

• \( F_g = mg \)

Where \( F_g \) is the gravitational force (or weight), \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \) on Earth. In the exercise, the gravitational force of sand poured into the container is needed to balance with the force exerted by the spring.
The required mass of sand was found to be about \( 10.62 \, \text{kg} \), which corresponds to the gravitational force balancing the hydraulic system's force requirements. Understanding gravitational force helps illuminate why a particular mass is necessary to create balance in a system based on weight.