Force Exerted by a Wedge Supporting a Sphere

[nautola]

Homework Statement

A solid sphere of radius R and mass M is placed in a wedge. The inner surfaces of the wedge are frictionless. Determine force exerted at point A and point B.

Homework Equations

ƩF=0
Ʃτ=0

The Attempt at a Solution

I'm pretty baffled about how to apply these equations to this situation. I would say to use some torque and the given angles to determine what force is applied. However, the forces are applied parallel to the distance traveled, so there is no torque from that.

 

[RTW69]

This is a statics problem. Don't worry about torque. The sphere isn't moving so sum of the forces in the x and y direction is 0. Let the center of the sphere be (0,0) you must find the coordinates for pt A and B. Draw a horizontal line through the origin (0,0). My guess is that the angle between the horizontal line and OB is beta and the angle between the horizontal line and OA is Alpha. Convince yourself of that by basic geometry relationships. Draw a free body of the sphere and sum of the forces in the x and y directions equal 0.

This problem is a little ugly with the geometry but doable.

 

[JHamm]

You need to have the upwards forces from A and B add to give the downwards force of gravity, also the horizontal components of the forces from A and B must be equal. There's two equations with two unknowns :)

 

[nautola]

Actually alpha and beta are the angles at the bottom. And I'm clueless as to how to use those angles to do the necessary trigonometry to find the components of A and B.

 

[JHamm]

The angle from the corner of the wedge to A to the center of the ball is \frac{\pi}{2} as is the angle from the corner to B to the center.

 

[RTW69]

I realize that alpha and beta are the angles at the bottom. Might they also be the angles from a horizontal line through the origin and OA and OB? You will need to do some geometry to confirm that. The coordinates for B are (Rcosβ, -Rsinβ) Do you know why? Same process for A.

 

[nautola]

The angle from the corner of the wedge to A to the center of the ball is \frac{\pi}{2} as is the angle from the corner to B to the center.

I thought that too. But sadly, alpha and beta aren't equal to each other.

 

[JHamm]

But the wedge is tangent to the ball at the points of contact so the angles are right angles.