"* The rotating blade of a blender tums with constant...

"* The rotating blade of a blender tums with constant angular acceleration $1.50 \mathrm{rad} / \mathrm{s}^{2}$. (a) How much time does it take to reach an angular velocity of $36.0 \mathrm{rad} / \mathrm{s}$, starting from rest? (b) Through how many revolutions does the blade turn in this time interval?

Key Concepts

Angular Displacement

Angular displacement refers to the angle through which an object rotates, often measured in radians. It provides a measure of the total rotation an object undergoes and is used together with angular velocity and acceleration in rotational motion analysis.

Angular Acceleration

Angular acceleration is the rate at which the angular velocity of an object changes with time. It plays a central role in rotational dynamics, analogous to linear acceleration in translational motion, and is typically measured in radians per second squared.

Angular Velocity

Angular velocity represents the rate of change of angular displacement with time. It quantifies how fast an object rotates and is measured in radians per second. Understanding angular velocity is essential for describing rotational motion.

Rotational Kinematics Equations

Rotational kinematics equations are the set of formulas that describe the relationships among angular displacement, angular velocity, angular acceleration, and time, mirroring their linear counterparts. These equations are fundamental in solving problems where rotation involves constant angular acceleration.

Conversion between Radians and Revolutions

Conversion between radians and revolutions is crucial when interpreting rotational measurements. Since one full revolution is equivalent to 2? radians, this conversion factor is used in problems where the result is desired in revolutions rather than radians.

Transcript

00:01 So we're looking at solving a question about angular acceleration.

00:07 So we have a rotating blade of a blender that turns with a constant angular acceleration of 1 .5 radiance per second squared.

00:19 We're trying to figure out how long does it take to move from rest, so that angular velocity of 0 to an angular velocity of 36 radiance per second.

00:30 So we know that the angular acceleration a is also equal to as the first derivative of the angular velocity with respect to time.

00:48 This also means that the change in the, with the constant acceleration, we know that the acceleration is also equal to the change in the velocity over.

01:03 The change in time between two points in time.

01:10 So therefore we can see that if we manipulate this equation and solve for time, we see that the amount of time will be equal to the change in velocity, angular velocity, divided by our acceleration.

01:36 All right.

01:37 So if we plug our numbers in, we see that we have our change in angular velocity, which is going from zero radiance per second to 36 radiance per second.

01:54 So that would be 36 radiance per second, right? and we'll divide that by a constant acceleration, which is 1 .5 radiance per second squared.

02:16 So if we solve this out, so we just take this.

02:19 And divide that by 1 .5, we get 24 seconds.

02:29 Okay? so we get 24 seconds.

02:32 So now part b of the question says through how many revolution, sorry, does the blade turn in this time interval? all right.

02:44 So if we continue on with our idea that the acceleration is equal, to the derivative of the velocity with respect to time.

03:00 We also know that if we take another derivative to this, we see that acceleration is also the double derivative respect to time of the angler displacement.

03:23 I call it theta...

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