Animated Solution for Physics - Laws of Motion: A smooth block is released at rest on a $45^{\circ }$ incline and then slides a distance $d$. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

The inclined plane is one of the most iconic setups in classical mechanics, famously used by Galileo to dilute the effects of gravity and study motion. In this problem, we are presented with a beautiful comparative experiment: a block sliding down an incline under two different conditions—one perfectly smooth, and one rough. By comparing the time it takes to cover the same distance, we can elegantly deduce the coefficient of kinetic friction.

Let's embark on this journey and break down the physics step by step.

Imagine a block of mass $m$ resting on an inclined plane that makes an angle $\theta =45^{\circ }$ with the horizontal. The primary engine driving the block down the slope is gravity. However, gravity acts straight down towards the center of the Earth. To understand the motion along the incline, we must resolve this gravitational force ($mg$) into two perpendicular components.

The component perpendicular to the surface is $mg\cos 45^{\circ }$. This force presses the block into the incline, and the surface pushes back with an equal and opposite Normal force, $N=mg\cos 45^{\circ }$.

The component parallel to the surface is $mg\sin 45^{\circ }$. This is the active force pulling the block down the slope.

Gravity Unopposed

In our first scenario, the incline is perfectly smooth. There is absolutely no friction to oppose the motion. The only force acting along the plane is the downward pull of gravity.

Using Newton's Second Law ($F_{net}=ma$), the acceleration $a_1$ of the block is simply:

Since the block starts from rest ($u=0$), we can use the second equation of kinematics to find the distance $d$ it covers in time $t_1$:

Enter Kinetic Friction

Now, let's reset the experiment, but this time, the surface is rough. As the block slides, kinetic friction ($f_k$) steps in, acting in the direction opposite to the motion. The magnitude of this frictional force is proportional to how hard the block is pressed against the surface:

This friction acts as a brake, reducing the net force pulling the block down. The new, reduced acceleration $a_2$ becomes:

Because the acceleration is lower, the block will naturally take a longer time, $t_2$, to cover the exact same distance $d$. The kinematic equation remains structurally identical:

Bridging the Two Worlds

Here is where the magic happens. Since the distance $d$ is identical in both scenarios, we can equate the two expressions we just derived:

Canceling the $\frac{1}{2}$ from both sides gives us a direct relationship between the accelerations and the times:

The problem provides a crucial constraint: the time taken on the rough incline is $n$ times the time taken on the smooth incline. Mathematically, $t_2=n{t}_1$. Substituting this into our equation yields:

The $t_1^2$ terms cancel out beautifully, leaving us with a profound and simple relation:

Unveiling the Friction Coefficient

Now, we substitute our expressions for $a_1$ and $a_2$ back into this master equation:

Notice how the acceleration due to gravity, $g$, is present in every term. This means the result is entirely independent of the planet you perform this experiment on! We can divide the entire equation by $g\cos 45^{\circ }$. This clever mathematical trick converts the sines into tangents:

Since we are dealing with a $45^{\circ }$ angle, we know that $\tan 45^{\circ }=1$. The equation collapses into a beautifully simple form:

All that remains is to isolate the coefficient of kinetic friction, ${\mu }_k$. Dividing both sides by $n^2$:

Rearranging the terms, we arrive at our final, elegant answer:

This result is deeply satisfying. It tells us that if $n=1$ (the times are the same), then ${\mu }_k=0$, which perfectly describes a smooth surface. As the rough surface becomes infinitely sticky ($n\to \infty$), the coefficient approaches $1$. Physics and mathematics in perfect harmony!