91Ó°ÊÓ

When objects are in motion against each other, kinetic friction plays a vital role. In this problem, where rocks are sliding uphill on an inclined plane, kinetic friction opposes their movement. It is calculated using the formula:

• \( f_k = \mu_k N \)
• \( N \) is the normal force, exerted perpendicular to the contact surface.
• \( \mu_k \) represents the coefficient of kinetic friction, which is 0.45 in this scenario.

The normal force in this case is influenced by gravity. Specifically, it is given as \( mg \cos \theta \), where \( m \) stands for mass and \( \theta \) is the angle of inclination (36 degrees here). Hence, kinetic friction can be framed as:

• \( f_k = \mu_k mg \cos \theta \)

This acts to decelerate the rocks as they move uphill. Understanding kinetic friction is essential since it impacts the net force and, consequently, the rocks' acceleration.

Static friction is crucial when determining if the rocks will remain stationary at the hill's highest point. Static friction prevents movement until the external force exceeds it.

• It is calculated as \( f_s = \mu_s N \), where \( \mu_s \) is the coefficient of static friction.
• In this context, \( \mu_s \) is 0.65.

To decide if the rocks stay put at the peak, compare the static friction with the downhill gravitational force component:

• The gravitational force working to pull the rocks down is \( mg \sin \theta \).
• If \( mg \sin \theta \leq \mu_s mg \cos \theta \), the rocks will stay.

For our problem, calculations reveal \( \tan 36^\circ < 0.65 \), signalling that static friction is not enough to prevent the rocks from sliding. Knowing when static friction can maintain equilibrium is invaluable in solving these physics problems.

Newton's Second Law of Motion is key to finding the acceleration of the rocks on the incline. It states that the acceleration \( a \) of an object is directly proportional to the net force \( F_{net} \) acting on it and inversely proportional to its mass \( m \):

• \( F_{net} = ma \)

For the rocks moving up the incline, the net force is determined by subtracting the kinetic friction (\( f_k \)) from the gravitational force component along the incline (\( mg \sin \theta \)). This derivation is expressed as:

• \( ma = mg \sin \theta - \mu_k mg \cos \theta \)
• By simplifying, acceleration can be expressed as \( a = g(\sin \theta - \mu_k \cos \theta) \)

This calculation assumes gravity is the dominating factor, compounded by the inclination. Newton’s law not only aids in determining acceleration but also offers a broader perspective on how objects move in response to varying forces.

Drawing free-body diagrams is an essential step in solving physics problems regarding forces. These diagrams illustrate all the forces acting on an object, like our rocks on an inclined plane.
To start, visualize and label each force:

• Gravity \( mg \), acting downwards through the center of mass.
• Normal force \( N \), perpendicular to the surface.
• Kinetic friction \( f_k \), opposing the motion of the rocks.

By breaking down the gravitational force into components, we understand better how each force influences the object:

• Component along the plane: \( mg \sin \theta \)
• Component perpendicular to the plane: \( mg \cos \theta \)

These forces collectively determine the movement of the rocks. Crafting a precise free-body diagram simplifies defining relationships between forces, eventually guiding how to apply formulas related to friction and Newton’s laws effectively.