Animated Solution for Physics - Kinematics: Two cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$, respectively. Their speeds are such that they make complete circles in the same time $t$. The ratio of their centripetal acceleration is

Two Cars, Two Tracks

Imagine you are standing at the center of two massive circular racing tracks. Two cars, with masses $m_1$ and $m_2$, are zooming around these tracks. The inner track has a radius $r_1$, and the outer track has a radius $r_2$.

At first glance, the problem throws a lot of variables at you: masses, radii, speeds, and time. But the secret to solving physics problems elegantly is knowing which variables actually matter and which ones are just there to distract you.

Same Time Period

The most critical piece of information in the problem is this: they make complete circles in the same time $t$.

In the language of circular motion, the time taken to complete one full revolution is called the Time Period ($T$). So, the problem is essentially telling us that $T_1=T_2=t$.

Why is this so powerful? Because the time period is directly linked to the angular velocity ($\omega$). Angular velocity tells us how fast an object is sweeping through angles, and it is defined as:

Since both cars take the exact same time to complete $2\pi$ radians (one full circle), their angular velocities must be identical. Therefore, ${\omega }_1={\omega }_2=\omega$.

Centripetal Acceleration

Now, we need to find the ratio of their centripetal accelerations. Centripetal acceleration is the inward acceleration that keeps an object moving in a circle.

There are two common ways to write the formula for centripetal acceleration: 1. In terms of linear speed ($v$): $a=\frac{v^2}{r}$ 2. In terms of angular speed ($\omega$): $a={\omega }^{2}r$

Which one should we use? Since we just established that $\omega$ is constant for both cars, the second formula is our golden ticket. It allows us to compare the accelerations directly without worrying about calculating their individual linear speeds.

Let's write down the acceleration for each car using our chosen formula.

For the first car on the inner track:

For the second car on the outer track:

Notice something interesting? The masses $m_1$ and $m_2$ are nowhere to be found! Centripetal acceleration is a purely kinematic quantity; it depends only on the geometry of the motion, not on the mass of the object experiencing it. The masses were a classic distractor.

Finally, we take the ratio of the two accelerations:

The ${\omega }^2$ terms cancel out perfectly, leaving us with a beautifully simple result:

The ratio of their centripetal accelerations is exactly equal to the ratio of their radii. The car on the larger track experiences a proportionally larger centripetal acceleration to maintain the same angular pace.