91Ó°ÊÓ

Kinetic friction is a crucial concept, especially when studying objects moving along surfaces like our skier on a slope. When an object slides across a surface, kinetic friction opposes its movement. It plays a vital role in determining how much force is needed to maintain that movement at a constant speed.
Kinetic friction can be calculated with the formula:

• \( F_f = \mu_k F_N \).

Here, \( F_f \) is the force of friction, \( \mu_k \) is the coefficient of kinetic friction, and \( F_N \) is the normal force.
In this scenario, the coefficient of kinetic friction is given as 0.120. This means that just 12% of the normal force acts in opposition to the skier's motion. Understanding these values helps us see how friction directly affects the force required by the tow bar to keep the skier moving uphill at a constant speed.

Gravitational force is an essential factor when dealing with inclined planes. It always acts downward towards the center of the Earth. On an incline, this force can be split into two components:
- **Parallel to the slope:** This component, represented as \( F_{g_{\text{parallel}}} \), pulls the object down the slope. It can be calculated as:

• \( F_{g_{\text{parallel}}} = mg \sin \theta \).

- **Perpendicular to the slope:** This component, noted as \( F_{g_{\text{perpendicular}}} \), acts perpendicular to the slope’s surface and is calculated by:

• \( F_{g_{\text{perpendicular}}} = mg \cos \theta \).

These components help in understanding how the gravitational force affects objects on slopes. In our exercise, \( F_{g_{\text{parallel}}} \) is one of the forces balanced by the tow bar to ensure the skier moves up the slope without acceleration.

Newton's First Law, often called the law of inertia, tells us that an object will stay at rest or move at a constant velocity unless acted on by a net external force.
For the skier, this means moving at the same velocity requires that the total forces parallel to the slope equal zero. Thus, the tow bar exerts a force equal to the combined effects of the parallel gravitational force and the kinetic friction.
This principle implies:

• Force from the tow bar (\( F_t \)) counteracts both kinetic friction and the force pulling the skier down the slope (gravitational force component parallel to the slope).

By applying Newton's First Law, we find that:

• \( F_t = F_f + F_{g_{\text{parallel}}} \).

Thus, this law helps us compute the particular force needed by the tow bar to keep the skier gliding smoothly up the incline.