91Ó°ÊÓ

Hooke's Law is an essential principle in physics that describes how springs behave. It tells us that the force needed to stretch or compress a spring by a certain distance is proportional to that distance. Mathematically, this is expressed as:\[ F = kx \]Where:

• \( F \) is the force applied to the spring (in Newtons).
• \( k \) is the spring constant (in N/m), indicating how stiff the spring is.
• \( x \) is the distance the spring is stretched or compressed from its equilibrium position (in meters).

In this problem, Hooke's Law helps determine how much our spring stretches when a block is pulled. With a spring constant \( k \) of 415 N/m, the problem involves finding the extent of stretching needed to exert a force of 99.54 N. This shows that the spring’s resistance is overcome leading to the exact amount of stretch calculated as 0.240 meters.

Newton's Second Law establishes the relationship between the net force acting on an object, its mass, and the acceleration produced. This is famously captured by the formula:\[ F = ma \]Where:

• \( F \) represents the net force applied to an object (in Newtons).
• \( m \) is the mass of the object (in kilograms).
• \( a \) is the acceleration of the object (in \/m/s²).

In our exercise, the second law is crucial to find how much force is generated by the acceleration of a 7.00 kg block with an acceleration of 14.22 m/s². Using this simple equation, the force exerted on the block by the spring and any driving factors is computed to be 99.54 N. This equates to the force applied by the spring, illustrating the universal applicability of Newton's Law in predicting an object's response to forces.

Constant acceleration indicates that an object's speed changes at a steady rate over time. This notion allows us to use a set of standard equations to predict the motion of objects subjected to uniform accelerations. The concept covers fundamental equations such as:

• Final velocity \( v = u + at \)
• Distance \( s = ut + \frac{1}{2}at^2 \)

In this scenario, we're informed that the block starts from rest. With these characteristics and a time frame of 0.750 seconds, we employed these equations to determine the block's movement and acceleration as it travelled 4 meters. This knowledge is pivotal to understanding how it linked to calculating the force required to stretch the spring.

Spring force refers to the force exerted by a spring upon any object that stretches or compresses it. As per Hooke's Law, the spring force is aligned with the product of the spring constant and the displacement from its rest position:\[ F = kx \]In our exercise, the spring is the instrument applying force to the block. With a spring constant \( k \) of 415 N/m and the force necessitated to move the 7 kg block as 99.54 N, the calculation shows how far the spring stretched from its natural position. Solving the equation, the spring force reveals a deformation of 0.240 meters. This step acknowledges that even though the movement is facilitated by constant acceleration, the spring's physical properties determine the precise details of the interaction involved.