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## Homework Statement

A uniform circular disc has mass M and diameter AB of length 4a. The disc rotates in a vertical plane about a fixed smooth axis perpendicular to the disc through the point D of AB where AD=a. The disc is released from rest with AB horizontal. (See attached diagram)

(a) Calculate the component parallel to AB of the force on the axis when AB makes an angle

*θ*with the downward vertical.

(b) Calculate the component perpendicular to AB of the force on the axis when AB makes an angle

*θ*with the downward vertical

## Homework Equations

KE = (1/2)Iω

^{2}

I of a disc about axis perpendicular to disc through centre is (1/2)mr

^{2}

Transverse component of acc. = r(dω/dt)

Radial component of acc. = rω

^{2}

## The Attempt at a Solution

I have done (a) okay with no problems and got the correct answer. I used a similar approach for part (b) but got the incorrect answer. For part (b):

The moment of inertia is (1/2)M(2a)

^{2}+ Ma

^{2}= 3Ma

^{2}

(using the parallel axis theorem).

Using conservation of energy:

(1/2)(3Ma

^{2})ω

^{2}= Mgacos

*θ*

(3/2)aω

^{2}= gcos

*θ*

Differentiating w.r.t.

*θ*gives:

3aω(dω/d

*θ*) = -gsin

*θ*

3aω(dω/dt)(dt/d

*θ*) = -gsin

*θ*

(dω/dt) = -(g/3a)sin

*θ**

Resolving forces: Mgsin

*θ*- Y = Ma(dω/dt)

Substituting in *: Mgsin

*θ*- Y = -(Mag/3a)sin

*θ*

Y = Mgsin

*θ*+ (1/3)Mgsin

*θ*= (4/3)Mgsin

*θ*

The answer is supposed to be (2/3)Mgsin

*θ*. I can't see what I have done wrong. Any help would be greatly appreciated!