91Ó°ÊÓ

Static friction is the force that keeps two surfaces at rest relative to each other. It needs to be overcome before an object starts sliding. In this exercise, the crate is held in place by static friction against the plane, preventing it from moving despite the gravitational pull. The magnitude of static friction is calculated using the formula:

• Static Friction: \( F_s = \mu_s \cdot N \)

where \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force acting on the object.
The normal force in this scenario is the perpendicular component of the crate's weight, given by \( N = m \cdot g \cdot \cos(\theta) \) where \( \theta \) is the angle of the plane.Static friction plays a critical role as it determines when motion will start, depending on whether the applied force exceeds this friction force.

Once the static friction is overcome, and the crate begins to move, kinetic friction takes over. Kinetic friction is usually less than static friction, making it easier for the object to keep moving once in motion. The kinetic friction force acting on the crate is calculated using:

• Kinetic Friction: \( F_k = \mu_k \cdot N \)

where \( \mu_k \) is the coefficient of kinetic friction. This force will act against the motion of the crate, requiring a continuous force to keep it moving.
In our problem, after determining the force needed to start the motion, we use kinetic friction to understand how it affects the acceleration and speed as the crate moves up the plane.

Gravitational force pulls the crate down towards the center of the Earth. On an inclined plane, this force can be split into components: one parallel to the plane and one perpendicular. The parallel component of gravitational force is what the static and kinetic frictions counteract:

• Gravitational Force Parallel: \( F_g = m \cdot g \cdot \sin(\theta) \)

where \( m \) is the mass of the crate, and \( g \) is the gravitational acceleration, typically \( 9.81 \, m/s^2 \). This component strives to slide the crate downwards, playing a crucial role in finding the net force acting on the moving crate.
Understanding gravitational components is essential to determining how much force needs to be applied to move the crate up against gravity.

Acceleration is the rate of change of velocity, crucial for understanding how quickly the crate speeds up once it starts moving. The net force applied to the crate, after overcoming both static and kinetic frictions, results in its acceleration. This acceleration is described by Newton's second law:

• Net Force: \( F_{net} = m \cdot a \)
• where \( a \) is acceleration.

In this exercise, once static friction is overcome, the remaining force from the applied force, minus the gravitational and kinetic friction forces, accelerates the crate up the plane.
Determining this acceleration helps calculate the time required to reach the desired speed of 2 m/s using the relation between force and motion. This ties together all forces involved to solve the problem comprehensively.