91Ó°ÊÓ

Differential equations are equations that relate a function to its derivatives. In this exercise, we use a differential equation to relate the net force acting on the box to its acceleration, using Newton's Second Law. This specific differential equation is formulated as \( F_{\text{net}} = m \cdot \frac{dv}{dt} \), where \( F_{\text{net}} \) is the net force, \( m \) is the mass of the box, and \( \frac{dv}{dt} \) is the derivative of velocity with respect to time, essentially the acceleration.
In our scenario, the net force \( F_{\text{net}} \) is the difference between the tension in the rope and the gravitational force, and this can be expressed as \( (36 \text{ N/s})t - mg \). Solving this differential equation involves integrating with respect to time to find the velocity function \( v(t) \).
By integrating, we find how the state of motion of the box changes over time, determining the velocity at different moments. This lays the groundwork for understanding how forces result in motion.

The force of gravity is a key player in the motion of objects, and it acts as a constant downward force on the box. Gravity force, often represented as \( F_g \), is calculated as the product of the mass of the object and the acceleration due to gravity. For our exercise, this is given by \( F_g = mg \), where \( m \) is the mass (3 kg) and \( g \) is the gravitational acceleration (approximately \( 9.8 \text{ m/s}^2 \)).
This downward force is countered by the tension in the rope. In our problem, understanding the gravity force helps us determine the net force, which dictates the acceleration according to Newton's Second Law.
It is essential to grasp that while the rope's tension changes over time, gravity remains constant. This constancy allows us to better understand how the box accelerates or decelerates as the situation progresses.

Tension is the force exerted along the rope that acts upward, opposing the gravitational force. For this exercise, the tension varies with time, given by the function \( T(t) = (36.0 \text{ N/s})t \). This means that the tension increases linearly with time, providing an increasing upward force on the box.
At the start, when \( t = 0 \), the tension is zero. As time passes, the tension grows stronger, reducing the net downward force on the box, potentially changing its direction of motion.
Knowing how the tension changes allows us to compute the net force at any given moment using \( F_{\text{net}} = T(t) - mg \), helping in the prediction of motion characteristics like velocity and position. Understanding tension is crucial since it's the counteracting force to gravity in this setup, and analyzing it helps in solving the exercise efficiently.