A problem on concurrent forces in a plane.

[Monsterboy]

Homework Statement

[/B]
A ball rests in a trough as shown in figure. Determine the angle of tilt $\theta$ with the horizontal so that the reactive force at B will be one-third at A if all surfaces are perfectly smooth.

2.Relevant equations

Application of Lami's theorem and some algebra and geometry

3.The attempt at a solution

I tried to derive an relation between $\theta$ and the Ratio between the reactions R_a and R_b ,so at $\theta$ =0⁰ the ratio between the reactions is one and at a particular $\theta$ the ratio becomes 3. The relation goes like this $f(\theta) = x\frac{Ra}{Rb}$ where $x$ is a constant
This didn't lead me anywhere because there are too many unknowns.

Another equation i can think of is $\theta$= x(R_a-R_b) so at $\theta =0$ , R_a =R_b at what $\theta$ will R_a=3R_b?? The required angle $\theta= x(2Rb)$ and $x= \frac{\theta}{2Rb}$

I don't see this going anywhere either.
Is this the right way to go ?

 

[SteamKing]

Homework Statement

°°[/B]
A ball rests in a trough as shown in figure. Determine the angle of tilt $\theta$ with the horizontal so that the reactive force at B will be one-third at A if all surfaces are perfectly smooth.

2.Relevant equations

Application of Lami's theorem and some algebra and geometry

3.The attempt at a solution

I tried to derive an relation between $\theta$ and the Ratio between the reactions R_a and R_b ,so at $\theta$ =0⁰ the ratio between the reactions is one and at a particular $\theta$ the ratio becomes 3. The relation goes like this $f(\theta) = x\frac{Ra}{Rb}$ where $x$ is a constant
This didn't lead me anywhere because there are too many unknowns.

Another equation i can think of is $\theta$= x(R_a-R_b) so at $\theta =0$ , R_a =R_b at what $\theta$ will R_a=3R_b?? The required angle $\theta= x(2Rb)$ and $x= \frac{\theta}{2Rb}$

I don't see this going anywhere either.
Is this the right way to go ?

I don't see why you can't draw a FBD of the ball and set up the normal forces on each contact surface using the conditions of static equilibrium.

Remember, the angle θ must be incorporated into the angle of the trough measured w.r.t. the bottom of the wedge. When θ = 0, then these angles will both be 30° as indicated on the diagram.

 

[Monsterboy]

Thanks ,i took a little more time than expected to incorporate the angle since i had to imagine the thing to rotate ,i got the answer 16.13 degrees ,is there anyway to derive an equation like the one i mentioned ? i am just trying to find some alternative methods( if there are any).

 

[SteamKing]

Thanks ,i took a little more time than expected to incorporate the angle since i had to imagine the thing to rotate ,i got the answer 16.13 degrees ,is there anyway to derive an equation like the one i mentioned ? i am just trying to find some alternative methods( if there are any).

I doubt it, since the answer, I think, depends on the trigonometry of the situation, which is not linear.