91Ó°ÊÓ

Understanding Newton's Second Law is fundamental in classical mechanics. This law states that the acceleration of an object depends on two things: the net force acting on that object and the object's mass. Mathematically, it can be expressed as

• \[ F = ma \]

Here, \( F \) represents the net force exerted on the object, \( m \) is the mass, and \( a \) is the acceleration. The force and acceleration are vector quantities, meaning they have both magnitude and direction.
In our exercise, the stone is subjected to the gravitational force (its weight) and the tension in the wire. Therefore, the net force acting on the stone is the difference between the gravitational force \( mg \) and the tension \( T \). This is crucial for determining the mass of the stone:

• \[ mg - T = ma \]

Thus, by rearranging this equation and incorporating the known acceleration of the stone, we can successfully solve for its mass. Remember: When objects accelerate, it's because there is an unbalanced force acting on them.

Kinematics is a branch of mechanics that describes the motion of objects without considering the causes of this motion. It focuses on parameters like displacement, speed, velocity, and acceleration. In our given problem, the stone's motion is analyzed using the kinematic equations.
One primary kinematic equation we use is:

• \[ \Delta s = v_0 t + \frac{1}{2} a t^2 \]

Here, \( \Delta s \) is the displacement, \( v_0 \) is the initial velocity, \( t \) is the time, and \( a \) is the acceleration. As the stone starts from rest, \( v_0 = 0 \), simplifying our equation to:

• \[ 12.6 = \frac{1}{2} a (3.00)^2 \]

Solving this gives us the stone's acceleration. Kinematics allows us to predict the future position and velocity of an object when its initial velocity, acceleration, and time duration of travel are known. It provides insights purely based on geometrical and time-based parameters of motion.

Torque plays a vital role in understanding rotational motion. It is similar to force in linear motion but applies to the rotational aspect. The torque \( \tau \) is calculated by

• \[ \tau = Tr \]

where \( T \) is the force applied (tension in the wire) and \( r \) is the radius of the pulley.
Rotational motion is also characterized by angular acceleration \( \alpha \), which relates to the linear acceleration of the stone by the equation

• \[ \alpha = \frac{a}{r} \]

The pulley's rotational inertia, \( I \), also influences its motion.
For a uniform disk like our pulley, the rotational inertia is given by

• \[ I = \frac{1}{2} M r^2 \]

where \( M \) is the mass of the pulley. Using the relationship between torque and angular acceleration, \( \tau = I\alpha \), we can find the tension in the wire as

• \[ T = \frac{1}{2} M a \]

This demonstrates how the linear and rotational properties of mechanics are intertwined, revealing the interconnected nature of motion.