# Angular velocity uniform rod problem

• Poutine
In summary, the conversation discusses a problem involving a turnstile rotating with initial angular velocity and a mud ball being thrown and sticking to one of the rods. The turnstile has a mass of M and length of d, while the mud ball has a mass of M/4 and initial speed of vi. The conversation goes on to discuss using the Conservation of Angular Momentum to solve the problem and finding the inertia of the turnstile to be (4/3)*M*d^2. It is also mentioned that the mud ball should be treated as a point mass with rotational inertia of I=mr^2. The conversation ends with a question about the angular momentum before contact, which is clarified to be Li=Ii&omega;i.

#### Poutine

Hi all, recently joined and having abit of trouble with a problem (several actually but I managed to figure out how to get started on one of them).

In any case the problem says:
Four thin, uniform rods each of mass M and length d = .75 m, are rigidly connected to a vertical axle to form a turnstile. The turnstile rotates clockwise about the axle, which is attached to a floor, with initial angular velocity w = -2.0 rad/s. A mud ball of mass m = M/4 and inital speed vi = 15 m/s is thrown and sticks to the end of one rod at an angle 60 degrees. Find the final angular velocity of the ball-turnstile system.

Now I think I have to use the Conservation of Angular momentum. As such I need to find the inertia of the turnstile which I think turns out to be IT=(4/3)*M*d^2

I think the formula I'm going end up with will probably be
It * wf + Ib * wf = IT* wi + angular momentum of the ball before the contact

The thing is I don't know how to get further from here.

I don't have a table of rotational inertias for different shapes in front of me, so I won't comment on that until tomorrow, after I look it up.

But as for where you go after that, you're basically done. You will have something like:

Ii&omega;i=If&omega;f

You know everything except &omega;f
(Don't forget to treat the mudball as a point mass, so Imud=mr2.)

Okay you answered my main question which is what is Imud? But I don't understand why that is true. Why is it mr^2, I was thinking it might have been 2/5 mr^2 but it seems I was wrong but I don't understand why.

Just to be certain the angular momentum before contact would be (M/4)*v*d*cos 60 right?

Originally posted by Poutine
Okay you answered my main question which is what is Imud? But I don't understand why that is true. Why is it mr^2, I was thinking it might have been 2/5 mr^2 but it seems I was wrong but I don't understand why.

They don't give you the radius of the ball of mud, so that tells you to treat it as a point (it must be very small compared to the length of the rods). The rotational inertia of a point is I=mr2.

Just to be certain the angular momentum before contact would be (M/4)*v*d*cos 60 right?

No, the cos(60) has nothing to do with it, as that only has to do with the configuration the system is in when the mud does make contact. The angular momentum beforehand is Li=Ii&omega;i.

## 1. What is the angular velocity uniform rod problem?

The angular velocity uniform rod problem involves determining the angular velocity of a uniform rod as it rotates around a fixed point or axis. This is a common problem in physics and engineering, and it requires knowledge of rotational motion and the properties of a uniform rod.

## 2. How is angular velocity defined?

Angular velocity is defined as the rate of change of angular displacement over time. It is typically measured in radians per second (rad/s) and is represented by the symbol ω. It is a vector quantity, meaning it has both magnitude and direction.

## 3. What are the key equations used to solve the angular velocity uniform rod problem?

The key equations used to solve this problem are the rotational kinematic equations, which include ω = Δθ/Δt, ω = ω0 + αt, and ω2 = ω02 + 2αΔθ. These equations relate the angular velocity (ω), initial angular velocity (ω0), angular acceleration (α), and angular displacement (θ).

## 4. How do you determine the angular velocity of a uniform rod?

To determine the angular velocity of a uniform rod, you first need to identify the fixed point or axis around which the rod is rotating. Then, you can use the key equations mentioned in question 3 to solve for ω. This typically involves knowing the initial angular velocity, angular acceleration, and/or angular displacement of the rod.

## 5. What factors can affect the angular velocity of a uniform rod?

The angular velocity of a uniform rod can be affected by several factors, including the length and mass of the rod, the force or torque applied to it, and the point or axis around which it is rotating. Friction and air resistance can also affect the angular velocity. In addition, the conservation of angular momentum can also play a role in determining the final angular velocity of the rod.