Angular velocity of object after being subjected to torque

[serverxeon]

In particular I am only looking at part c.

I know the simpler approach is to use energy.

But I am thinking if i can do an integration of all the torque throughout the process, and hence find the integration of the net angular acceleration and find the resultant change in angular velocity.

But I have no idea how to do the mathematics. Can anyone show me how the mathematics is done?

 

[tiny-tim]

hi serverxeon!

… if i can do an integration of all the torque throughout the process, and hence find the integration of the net angular acceleration and find the resultant change in angular velocity.

But I have no idea how to do the mathematics. Can anyone show me how the mathematics is done?

τ = Iα …

torque (of the weight) = 1/3 ML² d²θ/dt² ​

 

[serverxeon]

hi serverxeon!


τ = Iα …

torque (of the weight) = 1/3 ML² d²θ/dt² ​

hey thanks for the help.

sorry but I still don't know how to go on from here!

could you elaborate a bit further!

In particular I was looking at using the 'infinitesimal' dτ or dα approach.

 

[tiny-tim]

In particular I was looking at using the 'infinitesimal' dτ or dα approach.

there is no dτ or dα in this situation

use torque (of the weight) = 1/3 ML2 d²θ/dt² …

what is the torque? (as a function of θ)​

 

[Nickie]

I would first commend you on your critical thinking and exploration of different approaches to solving a problem. Both the energy approach and the integration approach have their merits and can lead to a better understanding of the system.

In regards to your question about how to perform the mathematics for the integration approach, it would depend on the specific system and the torque function. In general, the net torque can be calculated by adding up all the individual torques acting on the object. This can be represented by the equation:

Στ = Iα

Where Στ is the net torque, I is the moment of inertia of the object, and α is the angular acceleration. To integrate this equation, you would need to have a function for the torque as a function of time, and then use calculus to find the change in angular velocity over a certain time interval. This would involve finding the integral of the torque function with respect to time, and then using that value to solve for the change in angular velocity.

However, this approach may not always be practical or feasible, as it would require knowledge of the torque function and the ability to perform the necessary mathematical calculations. In many cases, the energy approach may be simpler and more straightforward. It would be important to consider the specific system and the desired level of accuracy in determining which approach would be more appropriate.

I hope this helps to clarify the mathematics behind the integration approach for finding the change in angular velocity after being subjected to torque. Keep exploring and questioning different methods to deepen your understanding of scientific concepts.