91Ó°ÊÓ

Static friction is the force that keeps an object stationary even when there is a force trying to move it. Imagine a heavy box on a ramp. It doesn’t start sliding immediately because static friction is acting on it. This force is directly related to two things: the normal force (the perpendicular force the surface exerts on the object) and the coefficient of static friction, which describes how "sticky" the surface is.
The formula for static friction at its maximum is given by:

• \( f_{s} = \mu_{s} N \)
• Where \( \mu_{s} = 0.35 \) in our exercise
• \( N = mg \cos \alpha \), where \( \alpha \) is the angle of the incline

Static friction must counteract any forces trying to make the object slide downward. But once the necessary force to overcome this friction is applied, the object begins to move, transitioning to kinetic friction.

Once an object begins moving, static friction switches to kinetic friction. Kinetic friction is usually less than static friction, which means that once an object starts sliding, it has less resistance to keep moving. In our example, once the box overcomes static friction on the inclined plane, it experiences kinetic friction instead.
The formula for kinetic friction is similar to static friction:

• \( f_{k} = \mu_{k} N \)
• Where \( \mu_{k} = 0.25 \) in the exercise

This means the box will continue to be affected by kinetic friction as it slides down the ramp. This force works against the motion of the box, slightly slowing it down but not enough to stop it. This is why kinetic friction impacts how quickly the box accelerates down the incline.

An inclined plane is a flat surface tilted at an angle, like a ramp. Such planes are often used to easily analyze forces. When an object, like our box, is placed on an inclined plane, gravity causes it to slide down. However, forces like friction are crucial in understanding its motion.
The angle of inclination determines the plane's steepness. Its effect on motion is described by breaking down gravitational force into two components:

• Parallel to the plane (\( mg \sin \alpha \)
• Perpendicular to the plane (\( mg \cos \alpha \)

When you adjust this angle \( \alpha \) in our exercise, it changes the balance of forces. At a specific angle, the component of gravity pulling the box downward becomes greater than the maximum static friction holding it in place, causing the box to start moving.

Acceleration refers to the rate at which an object speeds up or slows down. When the box begins to slide down the ramp, its acceleration is influenced by the forces acting on it. Primarily, the net force along the inclined plane surface determines the box's acceleration.
The net force accelerating the box downwards is the difference between the component of gravity pulling it downward and the kinetic friction opposing this pull. The formula is:

• \( a = g(\sin \alpha - \mu_{k} \cos \alpha) \)

For our box on a 19.3° inclined plane, this acceleration measures approximately 1.32 m/s². This value tells us how quickly the box picks up speed as it travels down the slope.

Kinematics deals with the motion of objects and how you calculate aspects like velocity and displacement over time. In this particular exercise, once we know the acceleration of the box as it slides, we can use kinematic equations to determine its final velocity after a certain distance.
For the box sliding 5 meters down the ramp, the initial velocity \( u \) is 0 since it starts from rest. Using the kinematic equation:

• \(v^2 = u^2 + 2as\)
• Where \( v \) is the final velocity
• \( a = 1.32 \, \text{m/s}^2 \)
• \( s = 5 \, \text{m} \)

Plugging in these values, the final velocity \( v \) comes out to approximately 3.63 m/s. This indicates how fast the box is moving after sliding the specified distance.