91Ó°ÊÓ

Understanding force components is crucial when analyzing how forces apply to objects. Forces can act in multiple directions, thus they have different components. A force component is a part of a force vector that acts along a particular direction, usually resolved into horizontal and vertical parts.
For instance, when a force is applied at an angle \(\theta\), the force can be broken down into two perpendicular components:

• **Horizontal Component (\(F_h\))**: This acts along the horizontal direction and is calculated using \(F_h = F \cos(\theta)\). The cosine function is used because it relates to the adjacent side of the angle in a right triangle.
• **Vertical Component (\(F_v\))**: This acts along the vertical direction and is determined by \(F_v = F \sin(\theta)\). The sine of \(\theta\) is used here as it corresponds with the opposite side relative to the angle.

As shown in the exercise, when \(\theta=0^{\circ}\), the whole force is horizontal, resulting in \(F_h = 100 \text{ N}\). At \(30^{\circ}\), the horizontal force becomes \(86.6 \text{ N}\), while at \(60^{\circ}\), it reduces to \(50 \text{ N}\). Recognizing these changes helps predict how an object moves based on varying angles of applied force.

Static friction acts to keep an object at rest when it is subjected to forces. It is the friction that needs to be overcome to start moving an object. The maximum static friction force is calculated as \(F_{s,\text{max}} = \mu_s F_N\), where \(\mu_s\) is the coefficient of static friction, and \(F_N\) is the normal force.
Details in the problem illustrate that when \(\theta=0^{\circ}\), static friction can support up to \(102.0 \text{ N}\) due to the \(100 \text{ N}\) applied force, preventing movement. At \(\theta=30^{\circ}\), the static friction drops to \(82.0 \text{ N}\), not enough to counter the \(86.6 \text{ N}\) horizontal force, causing the chair to slide. Conversely, at \(\theta=60^{\circ}\), the static friction of \(66.6 \text{ N}\) exceeds the \(50.0 \text{ N}\) force, keeping the chair stationary.
These dynamics highlight the importance of friction in determining an object's movement.

Normal force is the force exerted by a surface perpendicular to an object resting on it. It is crucial in maintaining equilibrium, as it counters the weight of the object. When extra vertical forces act, it adjusts accordingly.
The normal force \(F_N\) often balances gravitational force, calculated as \(F_N = mg\), where \(m\) is mass and \(g\) is gravitational acceleration. The added vertical component of external forces can reduce \(F_N\), as seen when angles like \(\theta = 30^{\circ}\) or \(60^{\circ}\) apply. Here, the additional vertical force is \(- F \sin(\theta)\), thus decreasing the normal force.

• **At \(\theta = 0^{\circ}\)**, no vertical force modifies the weight, so \(F_N = 245.25 \text{ N}\).
• **At \(30^{\circ}\)**, \(F_N\) decreases to \(195.25 \text{ N}\).
• **At \(60^{\circ}\)**, further reductions bring it to \(158.65 \text{ N}\).

This concept is vital since it directly influences static friction and stability, impacting whether the object slides or remains still.