Planar Kinematics of a Rigid Body

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Discussion Overview

The discussion revolves around a homework problem involving the planar kinematics of a rigid body, specifically focusing on the relationship between a driving shaft and a spur gear. Participants explore the implications of angular acceleration, initial angular velocity, and the geometric relationship between the shaft and gear.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a homework statement involving a motor rotating a shaft with a non-constant angular acceleration defined as α=0.06θ^2 rad/s^2, and seeks to determine the angular velocity of gear B after a specified angular displacement.
  • Another participant questions the relationship between shaft A and gear B, suggesting that additional information, such as a diagram, would be helpful for clarity.
  • Details about the dimensions of the driving shaft and gear B are provided, noting that shaft A has a radius of 12mm and gear B has a radius of 60mm, with shaft A turning gear B.
  • Clarification is made that the axes of shaft A and gear B are parallel but not co-axial, and that gear B is a spur gear driven by shaft A.
  • One participant emphasizes the importance of knowing the gearing ratio between shaft A and gear B to relate their angular motions effectively.
  • Another participant points out that the angular acceleration is not constant due to its dependence on θ, indicating that calculus may be necessary to find the angular velocity of the shaft.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the relationship between the shaft and gear, particularly the lack of information about the gearing ratio. There is also a recognition of the complexity introduced by the non-constant angular acceleration, suggesting that multiple approaches may be needed to solve the problem.

Contextual Notes

The discussion highlights limitations related to missing assumptions about the gearing ratio and the implications of non-constant angular acceleration, which may affect the solution process.

getty102
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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?
 
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What's the relationship between shaft A and gear B? Are there more gears? A picture would be nice.
 
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.
 
getty102 said:
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?
 
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.
 
getty102 said:
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.
 
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.
 
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.
 

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