A block is projected up an incline at angle \(\theta\). It returns to its initial position with half its initial speed. Show that the coefficient of kinetic friction is \(\mu_{\mathrm{k}}=\frac{3}{5} \tan \theta\)

Understanding the physics of an inclined plane is fundamental when dealing with problems in classical mechanics. An inclined plane is a flat surface tilted at an angle, \(\theta\), to the horizontal. This simple machine helps to lift objects by spreading the work over a longer distance. When an object is placed on an inclined plane, several forces come into play: the gravitational force, the normal force, and the force of friction.

The force of gravity can be broken down into two components: one that is perpendicular to the surface, which is counterbalanced by the normal force, and one that is parallel to the surface, which is the component that pulls the object down the plane. Friction is a resistive force that acts opposite to the direction of motion and is proportional to the normal force. It's important to note that the coefficient of kinetic friction, \(\mu_{\mathrm{k}}\), is a dimensionless quantity that represents the ratio of the force of kinetic friction to the normal force. By analyzing these forces, we can use Newton’s second law to predict the acceleration of the object along the inclined plane.

Newton's second law of motion is one of the cornerstones of classical mechanics and states that the force \(\vec{F}\) acting on an object is equal to the mass \(m\) of that object multiplied by its acceleration \(\vec{a}\): \[\vec{F} = m\vec{a}\]. In the context of an inclined plane, this law allows us to calculate the acceleration of a block sliding up or down the plane.

For the upward movement, the net force is the difference between the gravitational force pulling the block down the incline and the force of friction resisting the motion. In contrast, when the block moves downward, the gravitational force and the force of friction work in the same direction. By applying Newton’s second law to these situations, we obtain the equations of motion needed to describe the block’s behavior. Remember, the acceleration of the object will be different during its ascent and descent due to the opposite direction of the frictional force, and this is a critical aspect when solving related problems.

The principle of energy conservation states that in a closed system, with no external force doing work, the total energy remains constant over time. In our problem, two types of mechanical energy are relevant: potential energy due to the block's height and kinetic energy due to its motion.

When the block is projected up the incline and then slides back to its starting position, the total mechanical energy is not conserved because friction dissipates energy as heat. However, we can use the principle of energy conservation by considering the work done by friction. By equating the loss in kinetic energy on the way up to the work done against friction, and realizing that upon return, the block has half its initial kinetic energy, we can set up equations that reflect these energy transformations and ultimately solve for the coefficient of kinetic friction. The coefficient can then provide valuable insights into the interaction between the block and the surface of the incline.