(Mechanics) Pulley System Question

[koolitzb]

Homework Statement

Derive an expression for the angle alpha and determine the initial horizontal and vertical forces exerted at the point of attachment of the bell rope to the plank.
The system is in static equilibrium

Both ropes are light ie negligible mass.
The point of attachment has been circled in red on the diagram
The pulley wheel diameter may be considered as negligible with the pulley wheel central pin being at the same height as the initial height of the plank. the distance between the pulley pin and the end of the plank is 3000 mm

Distance a = 3000 mm, b = 0.75 x a mm

Homework Equations

The Attempt at a Solution

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Have tried resolving vertically and horizontally, but I believe there is not enough information to even establish an expression for alpha, yet alone the forces exerted at the point of attachment
Can someone please push me in the right direction as I have spent hours racking my brain.

 

[.Scott]

I don't think they are asking you to solve for alpha. Different alpha's will result in different plank accelerations.

 

[koolitzb]

Sorry, forgot to mention the system is in static equilibrium, at the initial stage

 

[jackwhirl]

Because a and b are known, you should be able to define alpha in terms of beta. Try drawing a dashed line up from the bell, and a second dashed line connecting the plank and pulley 3.

Or solve it as a side-angle-side? I can't remember the last time I did that...

 

[koolitzb]

Because a and b are known, you should be able to define alpha in terms of beta. Try drawing a dashed line up from the bell, and a second dashed line connecting the plank and pulley 3.

Or solve it as a side-angle-side? I can't remember the last time I did that...

I managed to get an expression for alpha in terms of beta and the distances, using the sine rule.
However I still do not see how it would be possible to find the components of the forces exerted at the point of attachment

Is it reasonable to assume the tension T2 in the second string (to the right) is equal to the 18kg x gravity (the force due to the weight on the end of the rope)
I did not believe it was but I can not find any other methods of working it out

 

[jackwhirl]

Is it reasonable to assume the tension T2 in the second string (to the right) is equal to the 18kg x gravity (the force due to the weight on the end of the rope)
I did not believe it was but I can not find any other methods of working it out

Given static equilibrium, that is a perfectly reasonable assumption. Otherwise, the mass would be accelerating.

 

[koolitzb]

Given static equilibrium, that is a perfectly reasonable assumption. Otherwise, the mass would be accelerating.

Ok, thank you.

Even given this assumption, I do not see enough information provided to be able to work out the forces exerted at the point of attachment

 

[jackwhirl]

What are your equations and unknowns?

 

[koolitzb]

What are your equations and unknowns?

The two equations found from resolving vertically and horizontally, T2 is now known as the weight 18kg, however alpha, beta and T1 are still three unknown

NEW
I can not form a correct third equation, I have tried using the geometry but something always seems to be missing

 

[jackwhirl]

I managed to get an expression for alpha in terms of beta and the distances, using the sine rule.

What about this equation?

 

[koolitzb]

What about this equation?

Sorry, forgot to mention I tried using that equation using the sine rule and it didnt work out due to the arcsin of the sin. Either way I don't believe this is the correct way of going about forming the equation from the geometry
That is the equation I ended up getting at the bottom labelled 1.

Note:
Apologies for the different values, but it is the actual concept of how to form the equation I am concerned about rather than the answer
Different handwriting as me and my friend are both attempting to work on this and are both completely stuck

 

[Merlin3189]

Can you take moments about your red-circled point for the b rope?
The wieght of the bell is a CW torque and the T₂ of the rope is an ACW torque. T₁ acts through the pivot, so can be ignored.

This looks a simple problem, but like you I'm struggling. Moments seems the only remaining option.

BTW in post #1, a=3 m and b= 0.75a = 2.25 m, but in post #11, a= 2.666 m and b = 2 m. I prefer the values in #1, if you can choose!