Animated Solution for Physics - Laws of Motion: A block of $200\text{ g}$ mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius $20\text{ cm}$. If the block takes $40\text{ s}$ to complete one round, the normal force by the side walls of the groove is

A Block in a Groove

Imagine a small block sliding smoothly inside a circular groove carved into a horizontal table. The block is moving at a constant speed, tracing out a perfect circle. But wait—Newton's First Law of Motion tells us that an object in motion wants to stay in motion in a straight line. So, why is the block turning?

It turns because the walls of the groove are physically forcing it to! As the block tries to move straight, it crashes into the outer vertical wall of the groove. The wall, in response, pushes back on the block. This inward push is the Normal Force ($N$), and it acts as the invisible tether keeping the block in its circular orbit.

Who Provides the Centripetal Force?

For any object to execute circular motion, it requires a net force directed towards the center of the circle. We call this the Centripetal Force ($F_c$). It is not a new fundamental force; rather, it is a role played by existing forces. In our scenario, the normal force from the outer wall plays this exact role.

Mathematically, the centripetal force required to keep an object of mass $m$ moving in a circle of radius $r$ with an angular velocity $\omega$ is given by:

Since the normal force $N$ is the only force acting in the radial direction, we can confidently state:

We also know that the angular velocity $\omega$ is related to the time period $T$ (the time taken to complete one full revolution) by the equation $\omega =\frac{2\pi }{T}$. Substituting this into our force equation gives us our master equation:

Crunching the Numbers

Before we plug in the numbers, we must ensure all our units are in the standard SI format (kilograms, meters, and seconds). This is a classic trap where many students lose marks!

Given values: - Mass, $m=200\text{ g}=0.2\text{ kg}$ - Radius, $r=20\text{ cm}=0.2\text{ m}$ - Time period, $T=40\text{ s}$

Now, let's substitute these pristine values into our master equation:

Let's simplify the term inside the bracket first. $\frac{2\pi }{40}$ reduces beautifully to $\frac{\pi }{20}$. Squaring this gives us $\frac{{\pi }^2}{400}$. Now, multiplying the mass and radius gives $0.2\times 0.2=0.04$.

Bringing it all together:

Using the standard approximation $\pi \approx 3.14$, we find that ${\pi }^2\approx 9.8596$. Therefore, the normal force is:

Scaling of Forces

Look at how incredibly small this force is! It is less than a thousandth of a Newton. Does this make physical sense? Absolutely. The block takes a full 40 seconds to complete a tiny circle of 20 cm radius. It is crawling at a snail's pace. Because it is moving so slowly, its inertia is easily overcome, and it requires barely any force to continuously change its direction.

However, notice the squared relationship in our master equation: $N\propto \frac{1}{T^2}$. If the block were to speed up and complete the circle in just 20 seconds (half the time), the required normal force wouldn't just double—it would quadruple! Always pay attention to how variables scale; it builds a deep, intuitive feel for physics.