Solve Rigid Body Problem: Uniform Disc Mass m Falls from Theta=60°

[Bucky]

Homework Statement

Uniform disc, mass m, falls from position where theta 60 degrees. (P is a fixed point on rim of disc).

Calculate the components of thrust along and perpendicular to GP on the smooth hinge at P when theta = 30degrees.

see attatched image for diagram

Homework Equations

The Attempt at a Solution

NB: theta with a dot is velocity (v), and theta with two dots is acceleration (r). Not sure if these are actually what the two of them mean, but that's what they'll be represented by.

Transverse Equation:

-ma(r)= mg sin theta - x
x = mgsin theta + ma(r) (1)


Radial Equation

ma (v)^2= y - mgcos theta
Y = ma (v)^2+ mgcos theta (2)

Rotational equation about P

I r= sum of moments
2/3 m(a)^2 r= -mgasin theta

2/3a r= -g sin theta

a r= -(3/2) g sin theta (3)

Integrating (3) gives

a r= -3/2g cos theta + c

c = 3/2g cos theta

at theta = 60

c = 3/2g cos 60
therefore
c = 3/4 g

so

a (v)^2= -3/2g cos theta + 3/4 g

at theta = 30, (3) gives

a r= -3/2g sin30
= -3/4g

(4) gives

a (v)^2= -3/2g cos30 + 3/4g
= -3/2 * root(3)/2g + 3/4g
= -3root(3)/4g + 3/4g

= (3-3root(3))/4 g


From (3), at theta = 60

-ma r= mgsin theta - x
-m(-g/3) = mgsin60 - x
1/3mg = root(3)/2mg - x
1/3mg = root(3)/2mg - x
2/6mg = 3root(3)/6mg - x
x = 3root(3) - 2/6 mg

which is wrong. So I won't even bother looking at my value for y...

And I'm not sure where I've went wrong :/ seems every time i attempt this question I get a different answer.

 

[Nate810]

Have I made a mistake in setting out the equations or something? Any help would be greatly appreciated!

 

[m2003]

There are a few issues with your solution attempt. First, the equations you have listed as "homework equations" do not seem to be relevant to the problem. I would recommend reviewing the relevant equations for solving rigid body problems, such as the equations of motion for a falling object and the equations for rotational motion.

Secondly, it is not clear what you are trying to solve for in your solution attempt. Are you trying to find the components of thrust along and perpendicular to GP, or are you trying to find the acceleration and velocity at a certain angle? It is important to clearly define what you are solving for before attempting the problem.

Additionally, it is not clear what the values of x and y represent in your solution attempt. It is important to define all variables and their units in order to accurately solve the problem.

I would recommend reviewing the relevant equations and clearly defining what you are solving for before attempting the problem again. It may also be helpful to draw a free body diagram to better understand the forces acting on the disc.