Planar Kinematics of a Rigid Body

[getty102]

Homework Statement

Due to an increase in power, the motor M rotates the shaft A with an angular acceleration(α=0.06θ^2) rad/s^2, where θ is in radians. If the shaft is initially turning at ω
=50 rad/s, determine the angular velocity of gear B after the shaft undergoes an angular displacement Δθ=10 revolutions.

Homework Equations

ω^2=ωsub0^2+2α(θ-θsub0)
v=ωr

The Attempt at a Solution

I converted 10 revs to 62.83 rad. I then used the first equation by subbing in ωsub0, α, and 62.83 for θ and θ-θsub0. I came up with 179.62rad/s for ωsubA. I then found the velocity of shaft A using v=ωr. vsubA=vsubB therefore I tried to find ωsubB using ω=v/r. This did not work. Any thoughts?

 

[gneill]

What's the relationship between shaft A and gear B? Are there more gears? A picture would be nice.

 

[getty102]

The driving shaft A has a radius of 12mm whereas gear B has a radius of 60mm. I don't have a picture but Shaft A is turning Gear B.

 

[gneill]

The driving shaft A has a radius of 12mm whereas gear B has a radius of 60mm. I don't have a picture but Shaft A is turning Gear B.

Gear B is mounted co-axially on shaft A?

 

[getty102]

The axis of both Shaft A and Gear B are parallel although not co-axial. Gear B is a spur gear driven by Shaft A.

 

[gneill]

The axis of both Shaft A and Gear B are parallel although not co-axial. Gear B is a spur gear driven by Shaft A.

Seems to me that this information is important to the solution of the problem. We need to know the gearing ratio between shaft A and gear B in order to relate their angular motions.

 

[getty102]

That information is not available. The only givens are: Drive shaft radius=12mm;Gear B radius=60mm;Drive shaft angular acceleration=0.06θ^2 rad/s^2; initial angular velocity of Drive shaft A= 50 rad/s. Find angular velocity of Gear B after 10revs of Drive shaft A.

 

[gneill]

It looks like the acceleration isn't constant. It's of the form α = k*θ², so finding the angular velocity of the shaft will require some calculus.