A body rotating with an angular speed of 600 rpm is uniformly accelerated to 1800 rpm in 10 s. The n

Q: Why don't we need to calculate the angular acceleration $\alpha$?

Because the acceleration is uniform, the average angular velocity is exactly halfway between the initial and final velocities. Using $\theta = \omega_{\text{avg}} \times t$ is a shortcut that bypasses the need for $\alpha$.

Q: What if I used the formula $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$?

You would get the exact same result! First, find $\alpha = \frac{\omega_f - \omega_0}{t} = \frac{30 - 10}{10} = 2 \text{ rev/s}^2$. Then, $\theta = 10(10) + \frac{1}{2}(2)(10)^2 = 100 + 100 = 200$ revolutions.

Q: Why did some books or solutions get 32 as the answer?

That is a classic unit confusion error! They calculated $\theta = 200$ but mistakenly assumed the unit was radians instead of revolutions. They then divided 200 by $2\pi$ to get $\approx 31.8$, which they rounded to 32. Always track your units carefully!

Q: Do I need to convert rpm to rad/s?

Not necessarily. If the question asks for the number of *rotations* (revolutions), it is much easier to work entirely in revolutions per second (rev/s). Converting to rad/s and then back to revolutions just adds extra steps and room for error.

Q: Is rpm a standard SI unit?

No, the SI unit for angular velocity is radians per second (rad/s). However, rpm (revolutions per minute) is widely used in engineering and everyday life, so you must be comfortable converting between them.

Question

A body rotating with an angular speed of $600 	ext{ rpm}$ is uniformly accelerated to $1800 	ext{ rpm}$ in $10 	ext{ s}$. The number of rotations made in the process is …… .

Sigma Percentile · Accepted Answer

### The Spin Cycle: Mastering Rotational Kinematics

Imagine you are watching a massive industrial flywheel spinning up. It starts at a brisk $600 	ext{ rpm}$ and, over the course of $10 	ext{ seconds}$, accelerates smoothly to a roaring $1800 	ext{ rpm}$. The question is simple but profound: exactly how many times did this wheel rotate during that $10$-second window?

This is a classic problem in rotational kinematics, and it perfectly mirrors the linear kinematics you already know. Let's break it down step-by-step and avoid a very common trap that catches many students off guard.

### The Unit Trap: RPM to RPS

In physics, units are everything. The problem gives us the angular speeds in **revolutions per minute (rpm)**, but the time is given in **seconds**. To make our units consistent, we must convert the speeds into **revolutions per second (rev/s)**.

Since there are $60 	ext{ seconds}$ in a minute, we simply divide by $60$:

$$\omega_0 = \frac{600}{60} = 10 	ext{ rev/s}$$

$$\omega_f = \frac{1800}{60} = 30 	ext{ rev/s}$$

This means the wheel starts by making $10$ full rotations every second and ends up making $30$ full rotations every second.

### The Shortcut: Average Angular Velocity

Because the wheel is *uniformly accelerated*, its speed increases at a constant rate. This unlocks a powerful shortcut: we can use the **average angular velocity**.

Just like in linear motion where $v_{	ext{avg}} = \frac{u + v}{2}$, in rotational motion:

$$\omega_{	ext{avg}} = \frac{\omega_0 + \omega_f}{2}$$

Plugging in our values:

$$\omega_{	ext{avg}} = \frac{10 + 30}{2} = 20 	ext{ rev/s}$$

On average, the wheel is making $20$ rotations every single second during this interval.

### The Final Calculation

To find the total number of rotations ($N$), we just multiply this average speed by the total time:

$$N = \omega_{	ext{avg}} 	imes t$$

$$N = 20 	ext{ rev/s} 	imes 10 	ext{ s} = 200 	ext{ rotations}$$

And there we have it! The wheel makes exactly $200$ rotations.

### The Infamous Mistake

If you look at some textbooks or online solutions, you might see an answer of $32$ for this exact problem. **This is mathematically incorrect!**

Here is what happens: students (and sometimes authors) calculate the angular displacement as $	heta = 200$, but they mistakenly assume the unit is **radians** instead of **revolutions**. They then divide $200$ by $2\pi$ to convert it to rotations, getting $\frac{200}{2\pi} \approx 31.8$, which they round to $32$.

**Do not fall for this!** Because we kept our angular velocity in **revolutions per second**, our resulting angular displacement is already in **revolutions**. There is absolutely no need to divide by $2\pi$. Trust your units, and trust the physics!

Animated Solution for Physics - Rotational Motion: A body rotating with an angular speed of $600 rpm$ is uniformly accelerated to $1800 rpm$ in $10 s$ . The number of rotations made in the process is …… .

Enter Numerical Value:

Visualized Solution

$Visualizing the Rotating Body$

$Kinematics of Rotational Motion$

$Unit Conversion$

$Calculating Average Angular Speed$

$Total Number of Rotations$

$Addressing a Common Pitfall$

The Sigma Insight: Kinematics of Rotational Motion

The Spin Cycle

The Unit Trap

The Shortcut

The Final Calculation

The Infamous Mistake

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