The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 s. The number of revolu

Q: Why do we need to convert rpm to rad/s?

Standard kinematic equations are derived using radians, which are dimensionless and mathematically consistent with linear kinematics. Using rpm directly in equations involving time in seconds will lead to incorrect results.

Q: Could we have kept the time in minutes instead of converting speeds to rad/s?

Yes! You could convert the time $t = 26 \text{ s}$ to minutes ($26/60 \text{ min}$) and work entirely in revolutions and minutes. This is actually a great shortcut!

Q: Why is the formula for number of revolutions $n = \theta / 2\pi$?

One complete circle (or revolution) corresponds to an angle of $360^\circ$, which is exactly $2\pi$ radians. Therefore, dividing the total radians by $2\pi$ gives the number of full circles completed.

Q: What does uniform acceleration mean in this context?

Uniform acceleration means the angular speed increases at a constant rate. This allows us to use the standard rotational kinematic equations, which are analogous to $v = u + at$ and $s = ut + \frac{1}{2}at^2$.

Q: Is there a way to solve this without finding angular acceleration first?

Absolutely! Since the acceleration is uniform, you can use the average angular velocity formula: $\theta = \left(\frac{\omega_0 + \omega}{2}\right) t$. This bypasses the need to calculate $\alpha$ entirely.

Question

The angular speed of truck wheel is increased from $900 	ext{ rpm}$ to $2460 	ext{ rpm}$ in $26 	ext{ s}$. The number of revolutions by the truck engine during this time is ……… .
(Assuming the acceleration to be uniform).

Sigma Percentile · Accepted Answer

### The Setup: Visualizing the Spinning Wheel

Imagine a massive truck wheel spinning on the highway. The driver steps on the gas, and the wheel's rotation speeds up from $900 	ext{ rpm}$ to $2460 	ext{ rpm}$ over a span of $26 	ext{ s}$.

We need to find out exactly how many times this wheel turned during this acceleration phase. This is a classic problem of **kinematics of rotational motion**, where the principles are identical to linear motion, just with a rotational twist!

### Step 1: Taming the Units

Before we apply any physics equations, we must ensure our units are consistent. Standard SI units for angular velocity are radians per second ($	ext{rad/s}$).

To convert revolutions per minute ($	ext{rpm}$) to radians per second, we multiply by $2\pi$ (since one revolution is $2\pi$ radians) and divide by $60$ (to convert minutes to seconds).

$$ \omega_0 = 900 	imes \frac{2\pi}{60} = 30\pi 	ext{ rad/s} $$

$$ \omega = 2460 	imes \frac{2\pi}{60} = 82\pi 	ext{ rad/s} $$

Now our initial and final angular velocities are ready for action.

### Step 2: Uncovering the Angular Acceleration

Since the problem states the acceleration is uniform, we can directly use the rotational equivalents of Newton's equations of motion. The first equation links angular velocities, acceleration, and time:

$$ \omega = \omega_0 + \alpha t $$

Let's plug our values into this equation to find the angular acceleration, $\alpha$.

$$ 82\pi = 30\pi + \alpha (26) $$

Subtracting $30\pi$ from both sides gives $52\pi$. Dividing by $26$, we find that the angular acceleration is exactly:

$$ \alpha = 2\pi 	ext{ rad/s}^2 $$

### Step 3: The Total Angular Displacement

Next, we need to find the total angle turned, $	heta$, during this time. We use the second equation of rotational motion:

$$ 	heta = \omega_0 t + \frac{1}{2}\alpha t^2 $$

Substituting our known values:

$$ 	heta = (30\pi)(26) + \frac{1}{2}(2\pi)(26)^2 $$

$$ 	heta = 780\pi + \pi(676) $$

Adding them up, the total angular displacement is:

$$ 	heta = 1456\pi 	ext{ rad} $$

### The Final Stretch: Counting the Revolutions

We have the total angle in radians, but the question asks for the number of revolutions ($n$).

Since one full revolution covers an angle of $2\pi$ radians, we simply divide our total $	heta$ by $2\pi$.

$$ n = \frac{	heta}{2\pi} = \frac{1456\pi}{2\pi} $$

$$ n = 728 $$

The truck engine made exactly **728 revolutions** during this time!

### The Pro-Tip: The Average Velocity Shortcut

As a pro-tip, you could have solved this much faster! For uniform acceleration, the average angular velocity is simply the arithmetic mean of the initial and final velocities.

By converting the speeds to revolutions per second ($	ext{rev/s}$), we get:

$$ f_0 = \frac{900}{60} = 15 	ext{ rev/s} $$

$$ f = \frac{2460}{60} = 41 	ext{ rev/s} $$

Their average is:

$$ f_{	ext{avg}} = \frac{15 + 41}{2} = 28 	ext{ rev/s} $$

Multiplying this average velocity by the time ($26 	ext{ s}$) gives us the total revolutions directly:

$$ n = 28 	imes 26 = 728 $$

Try this shortcut out, it's a massive time-saver for competitive exams!

Animated Solution for Physics - Rotational Motion: The angular speed of truck wheel is increased from $900 rpm$ to $2460 rpm$ in $26 s$ . The number of revolutions by the truck engine during this time is ……… . (Assuming the acceleration to be uniform).

Enter Numerical Value:

Visualized Solution

\text{Visualizing the Rotating Wheel}

\text{Unit Conversion: rpm to rad/s}

\text{Kinematic Equations for Rotation}

\text{Calculating Angular Acceleration } (\alpha)

\text{Calculating Angular Displacement } (\theta)

\text{Total Number of Revolutions } (n)

\text{Alternative Approach}

The Sigma Insight: Kinematics of Rotational Motion

The Setup

Step 1

Step 2

Step 3

The Final Stretch

The Pro-Tip

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