8.01SC | Fall 2016 | Undergraduate

# Classical Mechanics

## Week 11: Angular Momentum

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### Week 11 Problem Set

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Two identical particles form a system. At the instant shown in the figure the particles have equal and opposite momentums: $$\displaystyle p_2=- p_1=p$$.

(Part a) Determine a vector expression for the angular momentum of the system about the point A.

(Part b) Determine a vector expression for the angular momentum of the system about the point B.

(Part c) How do your results for angular momentum about A and B compare?

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Two identical particles of mass $$\displaystyle m$$ move in a circle of radius $$\displaystyle R$$, $$\displaystyle 180^ o$$ out of phase, at an angular velocity $$\displaystyle \vec{\omega } = \omega _ z\hat{k}$$ in a plane parallel to but a distance $$\displaystyle h$$ above the x-y plane. Treat the two particles as a system.

Calculate $$\displaystyle \vec{L}_ S$$, the angular momentum of the system about point $$\displaystyle S$$. Express your answer in terms of $$\displaystyle m$$, $$\displaystyle R$$, $$\displaystyle h$$, $$\displaystyle \omega _ z$$, $$\displaystyle \hat{k}$$, and $$\displaystyle \hat{r}$$ as needed.

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This is a summary of the main points from the lesson so far:

• When an object rotates about a fixed axis that is not the axis of symmetry (such as an axis passing through one end of an object), the angular momentum about a point on the axis depends on the location of the point along the axis.
• When an object rotates about a fixed axis that is not the axis of symmetry (such as an axis passing through one end of an object), the angular momentum is parallel to $$\displaystyle \vec{\omega }$$ only when calculated at the point of intersection between the axis of rotation and the object, $$\displaystyle \vec{L}_{intersect}=I_{intersect}\vec{\omega }$$.
• When a symmetric object rotates about the axis of symmetry, the angular momentum about a given point $$\displaystyle S$$ on the axis is $$\displaystyle \vec{L}_ S = I_{cm}\vec{\omega }$$, independent of the location of point $$\displaystyle S$$.

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A rigid ring of radius $$\displaystyle R$$ and mass $$\displaystyle m_1$$ is lying on a horizontal frictionless table and pivoted at the point $$\displaystyle P$$. The figure is an overhead view, gravity points into the screen. A point-like object of mass $$\displaystyle m_2$$ is moving to the right with speed $$\displaystyle v_ i$$. It collides and sticks to the ring at point $$\displaystyle A$$ on the ring as shown in the figure. After the collision, the particle sticks to the ring and both objects rotate together counterclockwise about the pivot point. The angular velocity is pointing out of the page with magnitude $$\displaystyle \omega _ f$$.

The goal of this problem is to find $$\displaystyle \omega _ f$$, the magnitude of the angular velocity of the ring-particle system after the collision.

(Part a) Consider the ring and the object as the system and the time interval that the collision lasts. Which of the following statements is true during the collision?

• The momentum is constant.
• The angular momentum about the pivot is constant. The mechanical energy is constant.

(Part b) What is $$\displaystyle I_{P}^{system}$$, the moment of inertia of the system of the ring with the point object stuck to it at point A, calculated about an axis passing through the pivot point $$\displaystyle P$$ and perpendicular to the plane of the ring? Write your answer in terms of $$\displaystyle m_1$$, $$\displaystyle m_2$$, and $$\displaystyle R$$.

(Part c) Calculate the angular momentum about the pivot point $$\displaystyle P$$ of the ring-particle system at the instants immediately before and after the collision. For direction, assume that $$\displaystyle \hat{i}$$ points to the right, $$\displaystyle \hat{j}$$ points up, and $$\displaystyle \hat{k}$$ points out of the page. Write your answer using some or all of the following: $$\displaystyle R$$, $$\displaystyle m_1$$, $$\displaystyle m_2$$, $$\displaystyle v_ i$$, $$\displaystyle \omega _ f$$, $$\displaystyle \hat i$$, $$\displaystyle \hat j$$, and $$\displaystyle \hat k$$.

(Part d) Determine an expression for $$\displaystyle \omega _ f$$, the angular speed of the system immediately after the collision. Write your answer using some or all of the following: $$\displaystyle R$$, $$\displaystyle m_1$$, $$\displaystyle m_2$$ and $$\displaystyle v_ i$$

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Problem Set 11 contains the following problems:

1. Bug Walking on Pivoted Ring
2. A Rigid Rod
3. Elastic Collision Between Ball and Pivoted Rod
4. Elastic Collision of Object and Pivoted Ring
5. A Spaceship and a Planet

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Fall 2016
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