Finding the time for an object to start rolling without slipping

AI Thread Summary
The discussion revolves around the torque and friction in the context of an object rolling without slipping. A participant questions the absence of a negative sign in the torque equation, suggesting it should reflect the direction of friction. Clarification is provided regarding the convention of positive velocity and rotation, explaining that for a disc rolling to the right, the angular acceleration is indeed negative. The relationship between linear velocity and angular velocity is also debated, with emphasis on the correct interpretation of signs based on the direction of motion. Overall, the conversation highlights the importance of understanding conventions in physics equations.
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Homework Statement
I am trying to derive the equation that my textbook presents (5.28), however, I notice that they don't use the right hand rule for the torque so there is a slight change in the sign in my derivation
Relevant Equations
##v_r = r\omega## for is a condition for rolling without slipping where ##v_r## is the speed of the COM of the object
For this,
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I don't understand why they don't have a negative sign as the torque to the friction should be negative. To my understanding, I think the equation 5.27 should be ##I\frac{d \omega}{dt} = -F_{friction}R## from the right hand rule assuming out of the page is positive.

Noting that ##f_k = \mu_kmg## and integrating both sides, I get the equation of motion ##\frac{-Rmg \mu_kt}{I} = \omega(t)##

I also get ##v(t) = v_0 - u_kgt##

So setting the two equations equal to each other in the relation for rolling motion:

##v(t) = R \omega (t) ##

I get ##t_r = \frac{v_0}{\mu_kg(1 - \frac{mR^2}{I})}##. Could someone please explain to me who is wrong and why?

Many thanks!
 
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ChiralSuperfields said:
assuming out of the page is positive.
What you quote of the problem does not indicate whether the motion is left to right or right to left. Assuming usual conventions and a positive velocity, it is L to R. In that case, the angular acceleration is clockwise, so negative and into the page.
Correspondingly, at rolling, ##v=-R\omega##.
 
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haruspex said:
What you quote of the problem does not indicate whether the motion is left to right or right to left. Assuming usual conventions and a positive velocity, it is L to R. In that case, the angular acceleration is clockwise, so negative and into the page.
Correspondingly, at rolling, ##v=-R\omega##.
Thank you for your reply @haruspex! However, do you please know where the equation ##v = -R \omega## came from? Sorry I have not seen that equation with the minus sign before (I have only seen ##v = R \omega##)

Many thanks!
 
ChiralSuperfields said:
Thank you for your reply @haruspex! However, do you please know where the equation ##v = -R \omega## came from? Sorry I have not seen that equation with the minus sign before (I have only seen ##v = R \omega##)

Many thanks!
The commonest convention is positive to the right for velocity and positive anticlockwise for rotation. If a disc is rolling to the right along a line underneath it then its rotation is clockwise, so a positive velocity means a negative rotation, etc.
You can think of it as being because the radius, measured from the axis to the ground contact is downwards, so negative. If it were rolling along the ceiling then there would be no minus sign.
 
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haruspex said:
The commonest convention is positive to the right for velocity and positive anticlockwise for rotation. If a disc is rolling to the right along a line underneath it then its rotation is clockwise, so a positive velocity means a negative rotation, etc.
You can think of it as being because the radius, measured from the axis to the ground contact is downwards, so negative. If it were rolling along the ceiling then there would be no minus sign.
Thank you for your help @haruspex! Your explanation makes sense
 
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