91Ó°ÊÓ

Understanding equilibrium conditions is crucial when solving problems involving static friction and inclined planes. In the context of the board leaning against a wall, equilibrium means that all forces and torques (rotational forces) are perfectly balanced so that the board remains stationary. This condition manifests in two primary ways: the net force and the net torque on the board must both be zero.

In simpler terms, the board does not move horizontally or vertically, and it does not rotate. For this to happen, the upward forces must equal the downward forces, and the right forces must equal the left forces. Additionally, any tendency for the board to rotate must be resisted by opposite torques. Thus:

• The sum of vertical forces must equal zero, which implies that the normal force from the ground must balance the gravitational force pulling the board down.
• The sum of horizontal forces must also be zero. Here, the frictional force preventing the board from sliding must be equal but opposite to any horizontal component of forces.

By recognizing these equilibrium conditions, students can derive specific equations needed to solve for unknown variables like the angle of inclination.

The normal force is a fundamental concept when dealing with objects at rest on a surface. It is the perpendicular force exerted by a surface to support the weight of an object resting on it. In the scenario with the leaning board, the normal force acts upwards from the ground, directly opposing the downward force of gravity.

The magnitude of the normal force becomes clear when considering equilibrium conditions. As the board is in equilibrium, we can deduce that the normal force ( N ) must counterbalance the vertical component of the board's weight ( W ext{cos}( heta) ). This relationship is essential for calculating other forces, like friction, and generally appears as:

• N = W ext{cos}( heta)

Normal force does not act alone but is part of a system of forces, including friction. Understanding how these forces work together can help predict whether the board will remain at rest or start to slide.

Frictional force is the force that opposes the relative motion or tendency to such motion of two surfaces in contact. For the board, the frictional force acts horizontally, preventing it from slipping along the ground.

In physics, the static frictional force can be calculated using the formula:
f = mu N Where mu represents the coefficient of static friction (0.650 in this problem), and N is the normal force. This relationship tells us that the frictional force depends directly on how strongly the surfaces are pressed together, characterized by the normal force, and the nature of the surfaces, represented by mu .

The frictional force must be sufficient to balance the horizontal component of the weight of the board, which acts to slide it downwards. By equating the frictional force to the horizontal force component, we derive:

• f = W ext{sin}( heta)

This relationship is crucial for determining the limiting angle theta at which the board will start to slip.

The angle of inclination, \theta , is the angle between the horizontal ground and the board. It plays a significant role in determining whether the board will slip or stay in place.

The gravitational force acting on the board can be split into two components relative to this angle: one acting parallel and the other perpendicular to the board. The angle affects both the normal force and the frictional force, making it a critical factor in stability. A crucial question arises: at what angle does the board begin to slide?

To find this smallest angle of inclination, one can set the expression for the frictional force equal to the horizontal component of the weight, as defined by the equilibrium conditions. Solving the equation:\[ \tan(\theta) = 0.650 \]

One can find:

• \theta = \tan^{-1}(0.650) \approx 33^{\circ}

This calculation tells us that any angle smaller than approximately 33 degrees will not supply the necessary frictional force to prevent the board from sliding.