Would you care to explain the meaning behind the differential values? I am confused because in a math course I have never seen a differential element used on its own like this, but it is all over engineering and physics courses. I understand it is meant to represent an "infinitesimal" value, but such isn't defined in mathematics and is considered non-rigorous. I am an engineering student btw, but we were never explicitly taught how we may or may not treat infinitesimals by our lecturers. What does it actually represent in equations such as this? Should I consider it just a very small value (ie. a shorthand for a delta approaching 0)? I understand the hand-waviness behind the differentials in physics and engineering is fine as long as you are dealing with single-variable calculus, but problems will arise without proper mathematical treatment of such elements in multivariable calc. (multivariable chain rule just to name one - can't simply cancel the differentials out as if they were parts of a fraction).
Don't we need to specify that the rope is massless? (that is an implicit assumption, right?) Or is there some reason that your infinitesimal free body diagram did not incorporate the gravitational force of mass?
In principle yes, but if the main goal is to transfer rotational motion from one place to another then a sprocket would probably be a better choice, as with the capstan you’d need to make sure that the coefficient of friction is large enough to ensure no slippage. The meaning of "large enough" will depend on how much torque will be applied to the system.
Dear sir, while writing the force balance equation along tangential and normal direction why didn't you take the contribution of the weight of the small element?
I'm just assuming that we can neglect the weight for modelling purposes. I can't remember whether or not it ends up being possible to solve the resulting differential equation exactly if the weight is included - might be an interesting exercise to give it a try. Also, a quick search suggests that the capstans that are actually used on boats are often oriented such that the rope is wound in the horizontal plane rather than the vertical plane. In this case, the diagram in the video would really be a top-down view and the weight would be acting into the screen, so a full 3D analysis would be required if we wanted to include the effect of weight!
Why we use this relation in band brake and rope pulley problem both while in band brake, pure slipping is there but in case of rope-pulley pure rolling is there between rope and pulley. This result is better suits for band brake(because pure slipping is there i.e maximum value of friction u×R is generated). Please help through video. Please. louuu from india.
Thank you very much for the explanation!.
Thanks for watching!
You're my hero
That's very kind! I'm glad you are enjoying the videos.
If (P not )is equal to zero the the required force would be zero so does this mean tha lt in such case looping the rope would be impossible?
Would you care to explain the meaning behind the differential values? I am confused because in a math course I have never seen a differential element used on its own like this, but it is all over engineering and physics courses. I understand it is meant to represent an "infinitesimal" value, but such isn't defined in mathematics and is considered non-rigorous. I am an engineering student btw, but we were never explicitly taught how we may or may not treat infinitesimals by our lecturers. What does it actually represent in equations such as this? Should I consider it just a very small value (ie. a shorthand for a delta approaching 0)? I understand the hand-waviness behind the differentials in physics and engineering is fine as long as you are dealing with single-variable calculus, but problems will arise without proper mathematical treatment of such elements in multivariable calc. (multivariable chain rule just to name one - can't simply cancel the differentials out as if they were parts of a fraction).
Don't we need to specify that the rope is massless? (that is an implicit assumption, right?) Or is there some reason that your infinitesimal free body diagram did not incorporate the gravitational force of mass?
You're right, we're just neglecting the weight!
@DrBenYelverton thank you for responding. And thank you for the video!
Is it possible to use this principle to replace chain drive and sprocket? How many wraps are optimal? Thank you
In principle yes, but if the main goal is to transfer rotational motion from one place to another then a sprocket would probably be a better choice, as with the capstan you’d need to make sure that the coefficient of friction is large enough to ensure no slippage. The meaning of "large enough" will depend on how much torque will be applied to the system.
Dear sir, while writing the force balance equation along tangential and normal direction why didn't you take the contribution of the weight of the small element?
I'm just assuming that we can neglect the weight for modelling purposes. I can't remember whether or not it ends up being possible to solve the resulting differential equation exactly if the weight is included - might be an interesting exercise to give it a try. Also, a quick search suggests that the capstans that are actually used on boats are often oriented such that the rope is wound in the horizontal plane rather than the vertical plane. In this case, the diagram in the video would really be a top-down view and the weight would be acting into the screen, so a full 3D analysis would be required if we wanted to include the effect of weight!
Why we use this relation in band brake and rope pulley problem both while in band brake, pure slipping is there but in case of rope-pulley pure rolling is there between rope and pulley.
This result is better suits for band brake(because pure slipping is there i.e maximum value of friction u×R is generated).
Please help through video. Please.
louuu from india.
Where did the e come from?
We got a natural log term by integrating the differential equation, so had to raise e to the power of both sides to remove the ln.