91Ó°ÊÓ

When objects like a package slide on surfaces, they often face resistance. This resistance is due to kinetic friction. Kinetic friction is a type of friction that acts when two surfaces move past each other. It always opposes the direction of motion.

The intensity of kinetic friction is determined by two main factors:

• The coefficient of kinetic friction (\( \mu_k \)), a number that represents how much friction two materials exhibit against each other.
• The normal force (\( N \)), which is the force perpendicular to the surfaces in contact.

To find the force of kinetic friction (\( f_k \)), we use the formula: \[ f_k = \mu_k \cdot N \]. Here, knowing that \( \mu_k = 0.40 \) and \( N = 70.72 \) N gives us \( f_k \approx 28.29 \) N. Kinetic friction does negative work on an object because it opposes its motion. Hence, when calculating work done, the angle between the force and the direction of motion is \( 180^\circ \). The calculated work done by kinetic friction is \( -56.58 \) J, indicating it lessens the energy available for motion.

An inclined plane is a flat surface tilted at an angle relative to the horizontal. It's one of the classic simple machines that makes moving objects up or down easier compared to lifting them vertically.

In our problem, the package slides down a chute tilted at \( 53.0^\circ \). This tilt alters the force of gravity, splitting it into two components:

• One acting perpendicular to the plane.
• Another acting parallel to the slope.

Gravity's component along the incline pulls the package downward. We calculate this force using \( F_g = mg\sin\theta \), where \( m = 12.0 \text{ kg} \), \( g = 9.8 \text{ m/s}^2 \), and \( \theta = 53.0^\circ \). This gives \( F_g \approx 94.08 \) N.
Tilted surfaces like inclined planes are crucial because they can change how forces act on objects. Here, gravity does \( 188.16 \) J of work, aiding the package's movement down the chute.

Net work refers to the total work done on an object as it moves through a force field. It's the sum of all individual works from different forces acting upon the object.

In this exercise, three forces interact with the package:

• Friction, which does -56.58 J of work.
• Gravity, contributing 188.16 J of work.
• The normal force, which does no work, i.e., 0 J, as it acts perpendicular to the direction of motion.

To find the net work (\( W_{\text{net}} \)), sum these contributions: \[ W_{\text{net}} = -56.58 + 188.16 + 0 = 131.58 \text{ J} \].Net work tells us the change in the object's kinetic energy as it moves down the incline. This means the package gains energy due to gravity's dominating influence over the negative work of friction. Net work's significance lies in showing how different forces combine to affect the object’s overall state of motion.