Vector Mechanics — Double Gear Rolling on a Rack

[Alexanddros81]

Homework Statement

The double gear shown rolls on the stationary lower rack; the velocity of its
center A is 1.2 m/s directed to the right. Determine (a) the angular velocity
of the gear, (b) the velocities of the upper rack R and of point D of the gear.

Relevant Equations

-

Hi!
My first question: How does he get the equation $\frac {x_A} {2πr_1} = -\frac {θ} {2π}$ ?

 

[BvU]

You forgot to define $\theta$ (Solution forgot it too !)
Ask youriself: how far does $A$ go to the right for one revolution ($\theta = 2\pi$) ? For half a revoluton ? For a given $\theta$ ?

 

[Chestermiller]

Homework Statement:: The double gear shown rolls on the stationary lower rack; the velocity of its
center A is 1.2 m/s directed to the right. Determine (a) the angular velocity
of the gear, (b) the velocities of the upper rack R and of point D of the gear.
Relevant Equations:: -

Hi!
My first question: How does he get the equation $\frac {x_A} {2πr_1} = -\frac {θ} {2π}$ ?

What is the definition of a radian? How is it connected to this question?

 

[bagasme]

What is the definition of a radian? How is it connected to this question?

A radian is ratio between length of the arc and its radius:

$$\theta = \frac {x} {r}$$

For a complete circle, it became:

$$\theta = \frac {2 \pi r} {r} \\
\theta = 2 \pi ~ \text {rad}$$

 

[Chestermiller]

A radian is ratio between length of the arc and its radius:

$$\theta = \frac {x} {r}$$

For a complete circle, it became:

$$\theta = \frac {2 \pi r} {r} \\
\theta = 2 \pi ~ \text {rad}$$

I was asking the OP to help him/her understand where the equation in question had come from.

 

[BvU]

But we got no reaction whatsoever from @Alexanddros81 -- not so nice !

 

[Alexanddros81]

Hi!
I will provide my thinking soon for this question

 

[Alexanddros81]

You forgot to define $\theta$ (Solution forgot it too !)
Ask youriself: how far does $A$ go to the right for one revolution ($\theta = 2\pi$) ? For half a revoluton ? For a given $\theta$ ?

for one revolution $A$ goes to the right $2\pi r_1$. For half a revolution $\pi r_1$.
For a given $\theta$ goes to the right $\theta r_1$?

 

[BvU]

For a given $\theta$ goes to the right $\theta r_1$

No. Theta is an angle. It does not go to the right. You mean: For a given $\theta$, A goes to the right $\theta r_1$. Correct.

Now think the lower rack out of the way. Le A be a fixed axis
Ask youriself: how far does B (rack R) go to the right wrt A for one revolution ($\theta=2\pi$) ? For half a revoluton ? For a given $\theta$ ?