Dynamics Physics - Acceleration Analysis / Gear Ratios /

[f2434427@rmqkr]

First time posting.

Homework Statement

Gears A and B, of Masses 4kg and 10kg, respectively, are rotating about their mass centers. The radius of gyration about the axis of rotation is 100mm for A and 300mm for B. A constant couple of C=0.75Nm acts on gear A. Neglecting friction, compute:

- Angular Acceleration of each gear
- Tangential contact force between the gears at CMay want to look at attachment for image.

Homework Equations

The Attempt at a Solution

Started with the smaller gear.

Using T=I$\alpha$ , where I=mr^2

Rearrange the equation to get $\alpha$ = 0.75 / (4*0.1^2) = 18.75 rad/s^2

Not to sure how to draw a relationship with the 2nd larger gear. Maybe using w1/w2 = r2/r1 then $\alpha$ = w^2 * r

Thanks,

 

[Simon Bridge]

Welcome to PF;
The applied torque has to accelerate both gears doesn't it?
If this were a linear case, you'd use a free body diagram to couple the masses wouldn't you?

 

[f2434427@rmqkr]

Ah yes, that would make sense. Forgot to account for the 2nd gears inertia

hmm. Dont quite understand what you mean by coupling the masses though

 

[Simon Bridge]

When you push on one mass, it pushes on the other one .. the masses are said to be coupled ... the motion of one affects the motion of another.

 

[f2434427@rmqkr]

Alrighty,

Well the force applied from gear A, will have an opposite reaction on Gear B.

T=F*d

F=0.75/0.15 = 5n

then applying this force to Gear B.

T=F*D = 5 * 0.45 = 2.25 Nm

alpha = T/I = 2.25/(10*0.3^2) =2.5 rad/s^2Looks about right to me... hopefully. Logically, first gear would have to spin faster than the 2nd.

 

[Simon Bridge]

That's right - the smaller gear will reach a higher angular velocity in the same time, so must have a higher acceleration. You should be able to see from the relative sizes how the accelerations are related.