91Ó°ÊÓ

Newton's Second Law is a fundamental concept in physics that explains how the motion of an object changes when it is subjected to external forces. This law is formulated as \( F = m \cdot a \), where \( F \) is the net force applied to an object, \( m \) is its mass, and \( a \) is the acceleration produced. In our exercise, we see this law in action when the 15 kg block undergoes acceleration.
The force needed to cause this acceleration can be calculated using the formula. Here, the acceleration \( a \) is found to be \( 10.0 \text{ m/s}^2 \).
This acceleration is achieved by applying a net force of \( 150 \text{ N} \), confirming Newton's assertion that the acceleration of an object is directly proportional to the net force and inversely proportional to its mass.
This fundamental understanding helps us solve the problem by dissecting the forces involved during the block's motion.

Hooke's Law gives us insight into how springs operate. It states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed, mathematically expressed as \( F = k \cdot x \).
Here, \( k \) is the spring constant, which measures the stiffness of the spring, and \( x \) is the distance of the stretch.
In the problem, the spring stretches differently in two scenarios: first by 0.200 m during acceleration and then by 0.0500 m when moving at constant speed.
Using Hooke’s Law, the spring forces during these stretches can be used to find the spring constant \( k \).

• During acceleration: \( F = k \cdot 0.200 \)
• At constant speed: \( F = k \cdot 0.0500 \)

By solving these equations, the spring constant is calculated to be \( 1000 \text{ N/m} \).
This reveals how knowing the stretch of the spring and the forces involved allows us to determine the spring's characteristics.

The coefficient of kinetic friction \( \mu_k \) is key to understanding how friction acts on moving objects. It is a dimensionless value that represents the frictional force between two surfaces in motion.
In this scenario, friction opposes the block's motion across the table. We can find the frictional force using the equation \( F_{\text{friction}} = \mu_k \cdot N \), where \( N \) is the normal force.

• Normal force \( N \) is the force perpendicular to the surface, calculated as \( 147.15 \text{ N} \) for the block.

The problem shows that with a spring extension of 0.0500 m, the frictional force is \( 50 \text{ N} \).
Using the relationship \( F_{\text{friction}} = \mu_k \cdot N \), we solve for \( \mu_k \) and find it to be approximately \( 0.340 \).
This highlights the role of friction during motion and how it can be quantified using fundamental physics principles.

Uniform acceleration occurs when an object’s velocity changes at a constant rate over time. This concept is crucial in analyzing the block's motion.
In this exercise, the block accelerates uniformly from rest to a velocity of 5.00 m/s over 0.500 s. The formula for calculating acceleration is given by \( a = \frac{v - u}{t} \), where \( v \) is the final velocity, \( u \) is the initial velocity, and \( t \) is the time taken.
Applying this to our problem, the acceleration \( a \) is determined to be \( 10.0 \text{ m/s}^2 \).
This consistent acceleration helps us compute the net force necessary using Newton's Second Law.
Understanding uniform acceleration allows us to predict an object's motion, ensuring precise calculation and aiding in solving complex dynamics problems.