- #1

Aurelius120

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- Homework Statement
- A uniform circular disc radius R and mass M with a particle of mass M fixed on edge. The system is free to rotate about chord PQ. It is allowed to fall from vertical. Find the linear speed of the particle when it reaches the lowest point.

- Relevant Equations
- ##P.E.=\int{Fdx}##

Question image:

The question should be solved by conservation of mechanical energy.( I assume surface density##\sigma## and acceleration due to gravity##g=const.##)Therefore:

$$PE_i+KE_i=PE_f+KE_f$$

The axis of rotation ##PQ## is line of zero potential. Then

Since coordinates of center are ##(0,\frac{R}{4})## I define the integral as:

$$PE_i=\int^{R}_{-R}\left( 2g \sigma \left(y+\frac{R}{4}\right) \sqrt{R^2-y^2}\right)dy +\frac{5MgR}{4}$$ y is with respect to center.

I thought I had it but then integration struck and I could not do it no more. Moreover the final answer keeps giving a ##pi##(wrong integration on my part I guess) term that does not cancel.

So my questions are :

The question should be solved by conservation of mechanical energy.( I assume surface density##\sigma## and acceleration due to gravity##g=const.##)Therefore:

$$PE_i+KE_i=PE_f+KE_f$$

The axis of rotation ##PQ## is line of zero potential. Then

**1)**##PE_i=\int Fdy##Since coordinates of center are ##(0,\frac{R}{4})## I define the integral as:

$$PE_i=\int^{R}_{-R}\left( 2g \sigma \left(y+\frac{R}{4}\right) \sqrt{R^2-y^2}\right)dy +\frac{5MgR}{4}$$ y is with respect to center.

**2)**$$KE_i=0$$**3)**$$PE_f=-PE_i$$**4)**$$KE_f=\frac{I\omega^2}{2}=\left( \frac{MR^2}{4}+\frac{MR^2}{16}+\frac{25MR^2}{16} \right)\frac{\omega^2}{2}$$I thought I had it but then integration struck and I could not do it no more. Moreover the final answer keeps giving a ##pi##(wrong integration on my part I guess) term that does not cancel.

So my questions are :

**1) How to find the potential energy of the system?**

2) Is there any other method that could give answers faster?

3) Finally is my integral correct? Any hint/help in further solving it?2) Is there any other method that could give answers faster?

3) Finally is my integral correct? Any hint/help in further solving it?

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