A machine part rotates at an angular speed of 0.060 rad / s ; its speed is then increased to 2.2

A machine part rotates at an angular speed of 0.060 rad / s...

A machine part rotates at an angular speed of $0.060 \mathrm{rad} / \mathrm{s} ;$ its speed is then increased to 2.2 $\mathrm{rad} / \mathrm{s}$ at an angular acceleration of $0.70 \mathrm{rad} / \mathrm{s}^{2} .$ (a) Find the angle through which the part rotates before reaching this final speed. (b) If both the initial and final angular speeds are doubled and the angular acceleration remains the same, by what factor is the angular displacement changed? Why?

Key Concepts

Angular Displacement

Angular displacement refers to the angle through which a rotating object turns over a period of time, typically measured in radians. It is a key outcome in rotational motion problems, representing the cumulative effect of angular speed and acceleration. Calculating angular displacement often involves using kinematic equations to integrate the effect of angular acceleration over time, accounting for changes in angular velocity.

Rotational Kinematics Equations

Rotational kinematics involves equations that describe the motion of rotating bodies under constant angular acceleration. These equations are analogous to linear kinematics and include relationships between angular displacement, initial and final angular speeds, angular acceleration, and time. For example, one fundamental equation is ?² = ??² + 2??, which relates the square of the final angular speed to the square of the initial angular speed plus twice the product of angular acceleration and angular displacement.

Scaling in Rotational Motion

Scaling in rotational motion examines how variations in angular speeds affect quantities like angular displacement when other parameters, such as angular acceleration, remain unchanged. By analyzing equations like ?² = ??² + 2??, it is possible to determine how doubling the initial and final speeds changes the angular displacement, since the displacement depends on the difference of the squares of the speeds. This concept is important in understanding how proportional changes in motion parameters impact the overall motion.

Transcript

00:01 Consider the case of a part on a machine rotating at an initial angular velocity of 0 .06 radiance per second.

00:09 We have omega -i for the initial angular velocity is 0 .060 radians per second.

00:19 This part experiences an angular acceleration alpha of 0 .70 radians per second squared, while it reaches its final velocity, omega f, of 2 .2 radians per second.

00:38 We are interested in finding out what angle the heart rotates through while it's going from its initial angular velocity to its final angular velocity.

00:49 And this will be delta theta.

00:54 We begin by writing down one of our equations for the rotational motion of a rigid object.

01:01 We have the final angular velocity.

01:03 Squared equals the initial angular velocity squared plus two times the angular acceleration times delta theta which is the change in the initial and final angles because we're interested in delta theta we'll rearrange this so that delta theta is on the left and everything else is on the right so we have delta theta equals the difference between the final angular velocity squared and the initial angular velocity squared, all over two times alpha the angular acceleration.

01:40 We already know that the final angular velocity is 2 .2 radiance per second.

01:46 So we have that.

01:47 We know that the initial angular velocity is 0 .06 radiance per second.

01:53 So we have that.

01:54 And we also know that the angular acceleration is 0 .7 radians per second squared.

02:00 And we also have that.

02:01 Therefore, we have everything we need to be able to solve for delta theta.

02:08 So we can plug these values in.

02:11 For omega f, we have 2 .2 radians per second, close the parentheses and square it, minus 0 .060 radians per second, close the parentheses and square that...

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