Determine the angular velocity and angular displacement

[joemama69]

Homework Statement

The pinion gear A has a constant angular acceleration of 3 rad/s^2. Determine the angular velocity and angular displacement of the output shaft C @ t=2s starting from rest. The shaft is fixed to B and turns w/ it.

A has a radius of 35mm, B has a radius of 125mm, C is unkhown radius

Homework Equations

just a note. when i use a = angular acceleration

The Attempt at a Solution

First I found the angular velocity at A
w(a) = w(0) + at where w(0) = 0, a = 3rad/s^2, t=2
=6rad/s
Next I transferred it to the bigger gear B
w(a)r(a) = w(b)r(b) where w(a) = 6rad/s, r(a) = 35mm, r(b) = 125mm
there fore w(b) = 1.68 ras/s
a(a)r(a) = a(b)r(b) where a(a) = 3rad/s^2, r(a) = 35mm, r(b) = 125mm
therefore a(b) = .84 ras/s^2

Since B & C are conected is a(b) = a(c).

I then tried to find the angular displacement @ c

a d(pheta) = w dw which gives .84pheta = .5w(b)^2
pheta = 1.68 rad.

But 1.68 rad is the angular velocity at B. Is this correct

 

[LowlyPion]

Looks to me like they are both 1.68 .

ω = a*t

θ = 1/2 a*t² or

2*θ = a*t*t

But when t = 2 that means

θ = 2*a = ω

 

[nlightner]

?

Yes, your solution is correct. To clarify, the angular displacement at C is indeed 1.68 radians, which is equal to the angular velocity at B. This is because the angular velocity at B and C are the same since they are connected and turn together. The angular displacement at C can also be determined by using the formula w = w0 + at, where w0 is the initial angular velocity at C (which is 0 since it starts from rest) and a is the angular acceleration at C (which is also 3 rad/s^2). Plugging in the values, we get w = 3(2) = 6 rad/s. Then, using the formula w = dw/dt, we can find the angular displacement at C by integrating w with respect to time from 0 to 2 seconds. This will give us the same result of 1.68 radians.