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Problem statementSolution video |

**DISCUSSION THREAD**

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**DISCUSSION and HINTS**

For your work on this problem, it is recommended that you use an observer attached to the disk. The observer/disk has two components of rotation:

- One component of
*ω*_{0}about the*fixed*-axis.**K** - The second component of
*ω*_{disk }about the*moving*-axis.**j**

Write out the angular velocity vector **ω** in terms of the two components described above.

Take a time derivative of **ω** to get the angular acceleration **α** of the observer/disk. When taking this derivative, you will need to find the time derivative of the unit vector * j*. How do you do this? Read back over Section 3.2 of the lecture book. There you will see:

*Acceleration of point A
*The motion of A is quite complicated. To better understand the motion of A, consider first the view of point A by our observer who is attached to the disk - what does this observer see in terms of relative velocity and relative acceleration: (

With this known relative motion, we can use the moving reference frame acceleration equation:

* a_{A}* =

In finding the acceleration of B, * a_{B}*, note that B moves with a constant speed on a circular path centered on point O. Use the path description to find

Problem statementSolution video |

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* CHANGE IN PROBLEM STATEMENT*: Please use

**DISCUSSION and HINTS**

For your work on this problem, it is recommended that you use an observer attached to the disk. The observer/disk has two components of rotation:

- One component of
*ω*_{0}about the*fixed*-axis.**J** - The second component of
*ω*_{1}_{ }about the*moving*-axis.**i**

Write out the angular velocity vector **ω** in terms of the two components described above.

Take a time derivative of **ω** to get the angular acceleration **α** of the observer/disk. When taking this derivative, you will need to find the time derivative of the unit vector * i*. How do you do this? Read back over Section 3.2 of the lecture book. There you will see:

*Velocty of point A
*The motion of A is quite complicated. To better understand the motion of A, consider first the view of point A by our observer who is attached to the disk - what does this observer see in terms of relative velocity: (

With this known relative motion, we can use the moving reference frame velocity equation:

* v_{A}* =

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**DISCUSSION and HINTS**

Bar OA has two components of rotation:

- One component of
*Ω*about the*fixed*-axis.**J** - The second component of
*θ*_dot about the*moving*-axis.**k**

Write out the angular velocity vector **ω** in terms of the two components described above.

Take a time derivative of **ω** to get the angular acceleration **α** of the bar. When taking this derivative, you will need to find the time derivative of the unit vector * k*. How do you do this? Read back over Section 3.2 of the lecture book. There you will see:

Problem statementSolution video |

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**Discussion and hints:**

The disk shown above has TWO components of rotation:

- a rotation rate of ω
_{0}about a fixed axis (the "+"*Y*-axis), and, - a rotation rate of ω
_{disk}about a moving axis (the "-"*x*-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the disk is given by:

* ω* =ω

The angular acceleration vector * α* is simply the time derivative of the angular velocity vector

- Recall that the
-axis is fixed. Since**J**is fixed, then d**J**/dt =**J**.**0** - Recall that the
-axis is NOT fixed. Knowing that, how do you find d**i**/dt?**i**

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**DISCUSSION and HINTS**

In this problem, we desire to relate the rotation rates of link AB and link OC. With the two rigid bodies connected by a pin-in-slot joint, we are not able to use the rigid body kinematics equations by themselves. Let's discuss that below.

*Velocity analysis
*Here, we can use the rigid body velocity equation to relate the motions of A and B:

**v**_{A} = **v**_{B} + **ω**_{AB} x **r**_{A/B}

However, we cannot use a rigid body velocity equation to relate the motion of points O and A (the reason for this is that O and A are not connected by a rigid body). In its place, we can use the moving reference frame velocity equation *with an observer attached to link OC*:

**v**_{A} = **v**_{O} + (**v**_{A/O})_{rel} + **ω** x **r**_{A/O}

where * ω* is the angular velocity of the observer, and

Combine these two equations to produce two scalar equations.

*Acceleration analysis
*We will use the same procedure for acceleration as we did for velocity - use a rigid body equation for AB and a moving reference frame equation relating O and A.

**a**_{A} = **a**_{B} + **α**_{AB} x **r**_{A/B} + **ω _{AB}** x (

where * α* is the angular acceleration of the observer, and

Combine these two equations to produce two scalar equations.

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**DISCUSSION and HINTS**

In this problem, we desire to relate the rotation rates of the slotted wheel and the disk. With the two rigid bodies connected by a pin-in-slot joint, we are not able to use the rigid body kinematics equations by themselves. Let's discuss that below.

Before we do, however, can you think of a good application for such a mechanism design? Take a look at this short video.

*Velocity analysis
*Here, we can use the rigid body velocity equation to relate the motions of P and C:

**v**_{P} = **v**_{C} + **ω**_{disk} x **r**_{P/C}

However, we cannot use a rigid body velocity equation to relate the motion of points O and P (the reason for this is that O and P are not connected by a rigid body). In its place, we can use the moving reference frame velocity equation *with an observer attached to the slotted wheel**:*

**v**_{P} = **v**_{O} + (**v**_{P/O})_{rel} + **ω** x **r**_{P/O}

where * ω* is the angular velocity of the observer, and

Combine these two equations to produce two scalar equations.

*Acceleration analysis
*We will use the same procedure for acceleration as we did for velocity - use a rigid body equation for the disk and a moving reference frame equation relating O and P:

**a**_{P} = **a**_{C} + **α**_{disk} x **r**_{P/C} + **ω _{disk}** x (

where * α* is the angular acceleration of the observer, and

Combine these two equations to produce two scalar equations.

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* DISCUSSION and HINTS*As can be seen from the animation below, the path taken by P is a bit complex. However, what if we put an observer on the non-extending section of the rotating arm - what motion would the observer see for P? Think about that.

Suppose that we do put an observer on the non-extending portion of the rotating arm. That observer would describe the motion of P as being on a straight path aligned with the *x*-axis. Specifically, we have:

*( v_{P/O})_{rel} *=

Using this observer, we can write down the velocity and acceleration of P using the moving reference frame velocity and acceleration equations:

**v**_{P} = **v**_{O} + (**v**_{P/O})_{rel} + **ω** x **r**_{P/O
}**a**_{P} = **a**_{O} + (**a**_{P/O})_{rel} + **α** x **r**_{P/O} + 2**ω** x (**v**_{P/O})_{rel} + **ω** x (**ω** x **r**_{P/O})

where * ω* and

Problem statementSolution video |

**DISCUSSION THREAD**

Ask and answer questions here. Either way, you will learn.

**DISCUSSION and HINTS**

In this problem, we desire to relate the rotation rates of link OA and the disk.

With the two rigid bodies connected by a pin-in-slot joint, we are not able to use the previous rigid body kinematics equations by themselves. Let's discuss that below.

*Velocity analysis
*Here, we can use the rigid body velocity equation to relate the motions of O and A:

**v**_{A} = **v**_{O} + **ω**_{OA} x **r**_{A/O}

However, we cannot use a rigid body velocity equation to relate the motion of points A and B (the reason for this is that A and B are not connected by a rigid body). In its place, we can use the moving reference frame velocity equation *with an observer attached to the disk*:

**v**_{A} = **v**_{B} + (**v**_{A/B})_{rel} + **ω** x **r**_{A/B}

where * ω* is the angular velocity of the observer, and

Combine these two equations to produce two scalar equations.

*Acceleration analysis
*We will use the same procedure for acceleration as we did for velocity - use a rigid body equation for OA and a moving reference frame equation relating A and B.

**a**_{A} = **a**_{O} + **α**_{AO}x **r**_{A/O} + **ω _{AO}** x (

where * α* is the angular acceleration of the observer, and

Combine these two equations to produce two scalar equations.