91Ó°ÊÓ

Newton's second law is a fundamental principle of physics that explains how the motion of an object changes when it is subject to forces. It is expressed with the equation \( F = ma \), where \( F \) is the net force acting on the object, \( m \) is the mass of the object, and \( a \) is the acceleration. In simpler terms, it states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

In our specific scenario with the inclined wooden plane and the sliding box, Newton's second law helps us understand why the box starts to move when the sheet is tilted to angle \( \theta_1 \), and why it moves at constant speed at angle \( \theta_2 \).

* For angle \( \theta_1 \), the force of static friction is exactly balanced by the component of gravity pulling the box down the plane.\[ f_s = mg \sin \theta_1 \]
* For angle \( \theta_2 \), the box is already moving, and the kinetic friction equals the parallel component of gravity since the box moves at a constant speed.\[ f_k = mg \sin \theta_2 \]
This beautiful application of Newton's second law allows us to relate the incline angles directly to friction coefficients.

Static friction is the force that keeps an object at rest when it is subjected to an external force. It prevents the object from sliding until the applied force exceeds a certain threshold.

In our exercise, static friction is what prevents the box from moving down the wooden sheet until it reaches the angle \( \theta_1 \). At this point, the force due to gravity just balances the maximum static friction force.

Key points about static friction:

• It is always equal and opposite to the force trying to cause motion up to a maximum point.
• The magnitude of static friction depends on the normal force \( N \) and the coefficient of static friction \( \mu_s \).
• The formula is \( f_s = \mu_s N \), where \( N = mg \cos \theta_1 \) is the normal force on an incline.

In our scenario, when the surface reaches \( \theta_1 \), we solve \( \mu_s \) using \( \mu_s = \tan \theta_1 \). This gives us the angle at which the static friction is overcome.

Kinetic friction acts on a moving object and opposes its motion. It comes into play once static friction is overcome, and the object starts sliding.

In the context of the exercise, once the angle is reduced to \( \theta_2 \), the box slides with constant speed. This indicates that the kinetic friction force perfectly balances the gravitational force pulling the box down the incline.

Important aspects of kinetic friction:

• Kinetic friction has a constant magnitude for a given system and depends on the normal force \( N \) and the coefficient of kinetic friction \( \mu_k \).
• Unlike static friction, kinetic friction remains constant regardless of speed.
• The equation for kinetic friction is \( f_k = \mu_k N \), where \( N = mg \cos \theta_2 \) for an inclined plane.

The problem simplifies to \( \mu_k = \tan \theta_2 \), showing that the angle directly provides the kinetic friction coefficient, explaining the constant speed of the box.