Calculating Kinetic Energy of a Segment: Where Did I Go Wrong?

[Saitama]

Homework Statement

(see attachment 1)

Homework Equations

The Attempt at a Solution

I have attached my working, the correct answer is (A) but I can't see where I went wrong.

Any help is appreciated!

 

[haruspex]

Your first line is wrong. How did you get 1+sin(θ)? You're measuring it down fron horizontal, so should be minus.
And in the last step, you've evaluated the integral of the sin(θ) part to zero. It won't be.

Another approach is to say the angular speed is v/R and work out the distance from the element to the point of contact.

 

[Saitama]

Your first line is wrong. How did you get 1+sin(θ)? You're measuring it down fron horizontal, so should be minus.
And in the last step, you've evaluated the integral of the sin(θ) part to zero. It won't be.

Oh yes, the integral sin(θ) won't be zero. I evaluated it correctly during the examination but did it wrong while posting up my attempt.
I did the necessary corrections as you said and got the right answer. One more question, would the limits for integration change from 0 to -pi?

Another approach is to say the angular speed is v/R and work out the distance from the element to the point of contact.

This approach is much easier. I am not much familiar with the instantaneous axis of rotation but this worked out pretty well. Thanks for the alternate method.

 

[haruspex]

One more question, would the limits for integration change from 0 to -pi?

No, you're measuring theta anticlockwise in the diagram from the "9 o'clock" position, so it will be 0 to pi.

 

[Saitama]

No, you're measuring theta anticlockwise in the diagram from the "9 o'clock" position, so it will be 0 to pi.

Got it, thanks haruspex!