Rotational Momentum of Rotating Rod with Masses

In summary, rotational momentum, also known as angular momentum, is the measure of an object's tendency to continue rotating about an axis. It is calculated by multiplying the object's mass, velocity, and distance from the axis of rotation. The moment of inertia, which is a measure of an object's resistance to changes in its rotational motion, can be calculated using the parallel axis theorem. Adding masses to a rotating rod will increase its rotational momentum and it has many real-world applications in physics, engineering, and sports.
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1. A rigid, massless rod has three particles with equal masses attached to it as shown below. The rod is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through the point P and is released from rest in the horizontal position at t = 0. Assume that m and d are known. (Use the following as necessary: m, d, and g.)

<..2d/3...>
O---------------O-------P----------O
<...d...><...d...>

(a) Find the moment of inertia of the system of three particles about the pivot.

(b) Find the torque acting on the system at t = 0.

(c) Find the angular acceleration of the system at t = 0.

(d) Find the linear acceleration of the particle labeled 3 at t = 0.

(e) Find the maximum kinetic energy of the system.

(f) Find the maximum angular speed attained by the rod. ωf =

(g) Find the maximum angular momentum of the system. Lf =

(h) Find the maximum speed attained by the particle labeled 2. vf =

2.

I was able to solve for (a), (b), (c), and (e). I need help with finding (d), (f), (g), and (h).

3. For (d) finding the linear acceleration, I started with
(torque of mass 3) = (I of mass 3)(alpha)

alpha = 2g/7d

a=r(alpha)

a= 4g/3 but this is incorrect

For part (f), I tried to use conservation of momentum. Li=Lf
(Ii)(wi) = (If)(wf)
It starts from rest so w initial is 0, so I could not solve using this way...

part (g) I think I can solve this if I can solve (f)

How do I approach (h)?

Homework Statement


Homework Equations


The Attempt at a Solution

 
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  • #2
For (d), you need to use the equation a = rα, where a is the linear acceleration, r is the distance from the pivot point to the particle, and α is the angular acceleration. In this case, r = d/3 since the particle is located at 2d/3 from the pivot and the rod has a length of d. So, a = (d/3)(2g/7d) = 2g/21. This is the linear acceleration of the particle labeled 3 at t = 0.

For (f), you can use the equation ωf = ωi + αt, where ωf is the final angular speed, ωi is the initial angular speed (which is 0 in this case), α is the angular acceleration, and t is the time. You already know the value of α from part (c) and you can find t by setting the equation for angular displacement (θ = ωit + 1/2αt^2) equal to π/2 (since the rod rotates 90 degrees). Solving for t gives t = √(π/α) = √(7d/2g). Substituting this into the first equation, you can solve for ωf to get ωf = αt = √(7g/d).

For (g), you can use the equation L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed. You already know the value of I from part (a) and you can find ω from part (f). So, Lf = (2/3)(md^2)(√(7g/d)) = (2/3)(m√(7d^3g)).

For (h), you can use the equation v = rω, where v is the linear speed, r is the distance from the pivot point to the particle, and ω is the angular speed. In this case, r = d/3 and you already know the value of ω from part (f). So, vf = (d/3)(√(7g/d)) = √(7gd/3).
 

1. What is rotational momentum?

Rotational momentum, also known as angular momentum, is the measure of an object's tendency to continue rotating about an axis. It is a vector quantity that depends on the object's mass, velocity, and distance from the axis of rotation.

2. How is rotational momentum calculated?

Rotational momentum is calculated by multiplying the object's mass, velocity, and distance from the axis of rotation. The equation is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

3. What is the moment of inertia?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the object's mass distribution and the axis of rotation. For a rotating rod with masses, the moment of inertia can be calculated using the parallel axis theorem.

4. How does adding masses to a rotating rod affect its rotational momentum?

Adding masses to a rotating rod will increase its rotational momentum. This is because the added masses increase the object's mass, which is a factor in the equation for rotational momentum. The distribution of the masses also affects the moment of inertia, which in turn affects the rotational momentum.

5. What are some real-world applications of rotational momentum?

Rotational momentum is a fundamental concept in physics and has many practical applications. It is used in designing vehicles, such as cars and planes, to ensure their stability and control. It is also used in the study of celestial bodies, such as planets and stars, to understand their motion and behavior. In sports, rotational momentum plays a crucial role in movements such as spinning, throwing, and swinging.

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