A piece of ice slides from rest down a rough \(33.0^{\circ}\) incline in twice the time it takes to slide down a frictionless \(33.0^{\circ}\) incline of the same length. Find the coefficient of kinetic friction between the ice and the rough incline.

When addressing a physics inclined plane problem, it's crucial to understand the unique forces at play. An inclined plane is a flat surface tilted at an angle, not parallel to the force of gravity. This setup divides the force of gravity into two components: one acting perpendicular to the plane and another parallel to it.

The equation of motion for an object sliding down a frictionless inclined plane is given by the formula:
\[d = 0.5g t^2 \sin(\theta)\] where \(d\) denotes the plane's length, \(g\) is the acceleration due to gravity (9.8 \(m/s^2\)), \(t\) is the time, and \(\theta\) is the incline's angle. In simpler terms, this equation helps us calculate how long it takes for an object to travel down the plane, given that no frictional forces are resisting its motion.

The frictional force is a force exerted by a surface as an object moves across it or makes an effort to move across it. In the context of our inclined ice block, friction opposes the gravitational component parallel to the incline's surface. The frictional force is calculated as the product of the normal force (perpendicular to the surface) and the coefficient of kinetic friction (\(k\)).

In the equation \(f_k = k \cdot N\), \(f_k\) represents the kinetic frictional force and \(N\) stands for the normal force, which, in the case of an inclined plane, equals \(mg \cos(\theta)\). The presence of frictional force causes the ice to take twice the time to slide down the rough incline compared to a frictionless one. This relationship helps to deduce the coefficient of kinetic friction by analyzing the impact on the object's acceleration and time of descent.

In many physics problems, including our ice on an incline, we use kinematics equations to describe the motion of objects. They connect displacement, acceleration, the time taken, and the initial and final velocity of an object. In the absence of air resistance, an object on an incline experiences constant acceleration, making these equations particularly helpful.

In the solution provided, two different sets of kinematics equations are used for the frictionless and rough incline scenarios. These are adapted to take into account the change in the net acceleration due to friction on the rough incline. For constant acceleration, a commonly used kinematic equation is: \[ d = v_i t + 0.5 a t^2 \] where \(v_i\) is the initial velocity (which, in our case, is 0 since the ice starts from rest), \(a\) is acceleration, and \(t\) is time. By rearranging these formulas, we can solve complex motion problems step by step.

The acceleration due to gravity, denoted as \(g\), is the acceleration gained by an object due to the gravitational pull of the Earth. Its standard value is approximately \(9.8 m/s^2\) near the Earth's surface. This acceleration impacts the motion of objects in freefall and, as in our case, objects moving along an inclined plane.

On an inclined plane, gravity's effect on the acceleration of an object is a function of the angle (\(\theta\)) of the incline. The parallel component of gravity that accelerates the object down the incline is given by \(g \sin(\theta)\), while the perpendicular component, which influences the normal force, is \(g \cos(\theta)\). Understanding how gravity works in tandem with other forces, like friction, allows us to solve for unknown variables such as the coefficient of kinetic friction, as seen in the original exercise.