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Angular velocity is a measure of how fast an object rotates or revolves. It tells you how much angle an object sweeps out in a unit of time. For circular motion, it is essentially the rate of change of angular displacement. The standard unit for measuring angular velocity is radians per second \(\left(\mathrm{rad}/\mathrm{s}\right)\).

• Angular velocity can be constant or change over time.
• It is analogous to linear velocity, but for rotational motion.

To calculate angular velocity, you can use the formula:\[\omega = \omega_0 + \alpha t\]where:

• \(\omega\) is the final angular velocity.
• \(\omega_0\) is the initial angular velocity (which is zero if starting from rest).
• \(\alpha\) is the angular acceleration.
• \(t\) is the time elapsed.

In the problem given, the blade reaches an angular velocity of \(36\,\mathrm{rad}/\mathrm{s}\) starting from rest over a specific time period. Calculating how long it takes can be done by rearranging the angular velocity formula to solve for the time, thus giving us the first step in handling such problems.

Revolutions refer to the complete turns around a center point or axis. When discussing angular motion, it tells us how many times an object goes full circle.

• A single revolution means moving a complete \(2\pi\) radians.
• It's a standard way to talk about full cycles or loops.

In the second part of the exercise, we calculate how many revolutions the blender blade completes in the given time interval while accelerating. Once we know the total angle it has rotated, in radians, we can easily find how many full cycles or revolutions it completes. This conversion uses the relationship between radians and revolutions. Consequently, it's vital to develop a sense of how one translates to the other.

By knowing the total angle in radians (calculated in the previous steps) and using the relationship that \(1\) revolution equals \(2\pi\) radians, you can find the number of revolutions:\[n = \frac{\theta}{2\pi}\]where:

• \(n\) is the number of revolutions.
• \(\theta\) is the angle in radians.

Having cut through the calculations, the blade makes approximately \(69\) revolutions.

Radians are the standard unit of angular measure in physics and engineering. Unlike degrees, which divide a circle into \(360\) equal parts, radians are defined so that a complete circle is \(2\pi\) radians.

• One radian represents the angle of a circle created when the arc length is equal to the radius.
• Radians provide a direct relationship between the linear and angular quantities.

The choice of using radians is practical: it naturally ties into mathematical formulations for circles and curves. This makes calculations involving angular velocity, and acceleration, much simpler.

Furthermore, almost every formula in rotational motion works more efficiently and elegantly with radians. For example, when computing the total angle turned (as required in our blender blade problem), using radians allows for direct use in kinetic equations:\[\theta = \frac{1}{2} \alpha t^2\]Here:

• \(\theta\) denotes the angular displacement in radians.
• \(\alpha\) is the angular acceleration.
• \(t\) is the time.

The problem shows the power of radians as they streamline the process of converting angular measurements to other forms, such as revolutions, strengthening the connection between linear and rotational dynamics.