Physics Dynamics: Angular and Linear Motion of Machine Components

[tom23]

Homework Statement

Gear ‘B’ on the machine shown below accelerates from rest to 250 rpm in 2 seconds. Determine the angular velocity and acceleration of A, and the linear acceleration and velocity of rack C. See image attached

Homework Equations

I don't know where i would start off with this question, help would be appreciated.

The Attempt at a Solution

 

[tiny-tim]

Welcome to PF!

Gear ‘B’ on the machine shown below accelerates from rest to 250 rpm in 2 seconds. Determine the angular velocity and acceleration of A, and the linear acceleration and velocity of rack C.

Hi tom23! Welcome to PF!

(how far apart are the teeth on B compared with A and C? )

First find the angular velocity and acceleration of B.

Then you know that the "tooth-rates" on A B and C are the same … eg if B rotates through 7 teeth of the cog, then both A and C also move through 7 teeth.

 

[tom23]

So I got this until now, is it right?

ang velocity = 250 rpm x 2 pi x (1/60) = 26.2 rad/s

ang acceleration = change in ang velocity / chang in time = 26.2 rad/s / 2s = 13.1 rad/s^2

Thus at B: tangential Velocity at B = ang velocity*radius1 = 26.2*20 = 524 cm/s

tangential Acceleration at B = ang acceler.*radius1 = 13.1*20 = 262 cm/s^2

Therefore on A:

Angul. Velocity at A = tang. velocity/radius2 = 524/15 = 34.9 rad/s
Angul. Acceler at A = tang. acceler./radius2 = 262/15 = 17.5 rad/s

Does this two answers sound right, now how would i go on to solve for linear accel. and velocity of rack C?

 

[tiny-tim]

So I got this until now, is it right?

ang velocity = 250 rpm x 2 pi x (1/60) = 26.2 rad/s

ang acceleration = change in ang velocity / chang in time = 26.2 rad/s / 2s = 13.1 rad/s^2

Thus at B: tangential Velocity at B = ang velocity*radius1 = 26.2*20 = 524 cm/s

tangential Acceleration at B = ang acceler.*radius1 = 13.1*20 = 262 cm/s^2

Yes, that's fine.

Therefore on A:

Angul. Velocity at A = tang. velocity/radius2 = 524/15 = 34.9 rad/s
Angul. Acceler at A = tang. acceler./radius2 = 262/15 = 17.5 rad/s

You haven't answered my question: how far apart are the teeth on B compared with A and C?

If they're the same distance apart, then your answer is right.

now how would i go on to solve for linear accel. and velocity of rack C?

Same method … you have the tangential velocity of A, so you can find the linear velocity of C.

 

[tom23]

Tey didnt give the distance, so i believe it is the same, so for linear velocit and accel at C, i got:

V = 34.9 (30cm) = 1047 cm/s
a = 17.5 (30cm) = 525 cm/s

Does it look right?

Thanlks for all your help above!

 

[tiny-tim]

Hi tom23!

Tey didnt give the distance, so i believe it is the same, so for linear velocit and accel at C, i got:

V = 34.9 (30cm) = 1047 cm/s
a = 17.5 (30cm) = 525 cm/s

Does it look right?

Thanlks for all your help above!

Yes … that's fine … if the teeth are all the same distance apart, then the tangential velocities and the linear velocity are all the same.

(though I'd have done it directly from B … 26.2(40), rather than from A, as you did! )