A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s 2 . Calculate its angular speed 8 s after starting to accelerate. BONUS:Calculate its linear displacement in the 8 s.

Hey, guys. Let's check out this practice problem. Here we have a tiny object that spins with 5 radians per second around a circular path. So tiny objects spins around a circular path. So the radius of the path is 10 meters and the object spins obviously at the edge of this circle. So the distance from the center, little r, is the same as the radius of the circle, 10 meters, and it spins with a omega of 5 radians per second. It's a tiny object, so it doesn't have radius. And it says here that the object accelerates at 3 radius per second square, so alpha equals 3. This is therefore our omega initial. And we want to know its angular speed 8 seconds after starting to accelerate. So I want to know Omega final. Oops. I want to know Omega final and the time, delta t, time elapsed is 8 seconds. Okay. So these are a bunch of motion variables. So we're going to use one of the motion equations. The 5th variable that's missing here is delta theta. That's my my ignored variable. Put a little sad face. I got 1, 2, 3 things I can solve this. I'm going to use the only equation that doesn't have, delta theta in it, which is the first one. So Omega final equals Omega initial plus alpha t, and we're looking for omega final. So omega final equals 5+ alpha is 3 and time is 8. So this is going to be 29 radians per second. Very straightforward. Now for the bonus question, it says calculate its linear displacement in those 8 seconds. Linear displacement is delta x. Now Delta x is obviously not here. Okay. But Delta x is linked, or it can be converted to from delta theta. So what we're going to do is we're going to find delta theta first because remember that delta x is r delta theta. So to find delta theta, I already know that this is 29. So I know 4 out of the 5 variables, which means I can use any equation that has a delta theta in it. I'm going to use the third one because in the third one, delta theta is by itself on the left. So it's ready to go. Omega initial t plus half of alpha t squared. Let's plug in the numbers. The initial velocity is 5 times 8. The acceleration is 3 and time is 8. When you plug all this in, you get a 136 and the unit is obviously radians because remember these equations, we get radians as the answer, okay, as the unit 4 delta theta. All right. Cool. So I got this. I can now plug this into delta x. Delta x is r delta theta. R is the distance, the radial distance, which is 10 meters. So I got 10 136 radians and I get 1360. We're looking at Delta x. Therefore, the answer, the unit must be meters. Alright. So these are the 2 pieces. Hopefully you got this right. Pretty straightforward. Let me know if you have any questions.

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Equations of Rotational Motion

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Multiple Choice

A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s². Calculate its angular speed 8 s after starting to accelerate.
BONUS:Calculate its linear displacement in the 8 s.

19 rad/s

29 rad/s

43 rad/s

53 rad/s

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Verified step by step guidance

Identify the given values: initial angular speed \( \omega_0 = 5 \text{ rad/s} \), angular acceleration \( \alpha = 3 \text{ rad/s}^2 \), and time \( t = 8 \text{ s} \).

Use the formula for angular speed under constant angular acceleration: \( \omega = \omega_0 + \alpha t \).

Substitute the given values into the formula: \( \omega = 5 \text{ rad/s} + (3 \text{ rad/s}^2 \times 8 \text{ s}) \).

To find the linear displacement, use the formula for angular displacement \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).

Convert the angular displacement to linear displacement using the relationship \( s = r \theta \), where \( r = 10 \text{ m} \) is the radius of the circular path.