Engine. A car engine accelerates from 1000 to 2000 rpm at a constant rate during 15 s. (a) Find the angular acceleration of the engine. (b) Find the number of rotations the engine revolves from it starts at 1000 rpm until it has accelerated to 2000 rpm.

Angular acceleration refers to the rate at which the angular velocity of an object changes with time. It is denoted by \(\alpha\). In this context, a car engine's angular acceleration can be calculated using the formula \(\alpha = \frac{\omega_f - \omega_0}{t}\). Here, \(\omega_f\) and \(\omega_0\) are the final and initial angular velocities respectively, and \(t\) is the time interval. To find the angular acceleration, simply subtract the initial angular velocity from the final angular velocity and divide by the time interval. The resulting value is given in radians per second squared (rad/s²).

Angular velocity is a measure of the rate of rotation. It tells us how fast something is spinning and is usually represented by the symbol \(\omega\). The units of angular velocity are radians per second (rad/s). For the car engine example, we need to convert the speeds given in revolutions per minute (RPM) into rad/s. The conversion factor used is \( 1 \text{ RPM} = \frac{2\pi \text{ rad}}{60 \text{ s}} \). Using this conversion, we can transform the initial 1000 RPM and final 2000 RPM into their respective rad/s values to make further calculations easier and more standardized.

In problems involving angular mechanics, it is often necessary to convert between different units of measurement. Common conversions include RPM (revolutions per minute) to rad/s (radians per second), and radians to rotations. These conversions enable easier manipulation and consistency of units when performing calculations. For example, converting RPM to rad/s uses the relationship: \(1 \text{ RPM} = \frac{2\pi \text{ rad}}{60 \text{ s}}\). Another key conversion is between radians and rotations where \(1 \text{ rotation} = 2\pi \text{ rad}\).

To solve angular mechanics problems, you'll frequently need to convert RPM to radians per second. For instance, the initial angular velocity in the car engine example is 1000 RPM. Using the conversion formula \( 1 \text{ RPM} = \frac{2\pi \text{ rad}}{60 \text{ s}} \), we calculate: \( 1000 \times \frac{2\pi}{60} \approx 104.72 \text{ rad/s} \). Similarly, the final angular velocity of 2000 RPM converts to \( 2000 \times \frac{2\pi}{60} \approx 209.44 \text{ rad/s} \). This conversion is crucial for accurate calculations in angular mechanics.

Angular displacement, denoted as \(\theta\), measures the angle through which an object rotates. It is the rotational equivalent to linear displacement. You can find angular displacement using the formula \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\). Let's break this down: \(\omega_0\) is the initial angular velocity, \(t\) is the time interval, and \(\alpha\) is the angular acceleration. For the car engine example, \(\theta\) can be calculated by first finding the contributions from both the initial angular velocity and the acceleration over time. This displacement is found in radians, which can then be converted to rotations using \(1 \text{ rotation} = 2\pi \text{ rad}\).