Circles in a Box: Forces on Uniform Spheres at 35* Angle in Fixed Container

In summary, two identical uniform spheres, each weighing 75 N, are at rest on the bottom of a fixed rectangular container. The line of centers of the spheres makes an angle of 35 degrees with the horizontal. The forces exerted on the spheres by the container bottom, the container sides, and the force that each sphere exerts on the other are all calculated. The container bottom exerts a force of 150 Newtons on the spheres, while the container sides exert a force of 107 Newtons on each sphere. The spheres also exert a force of 131 Newtons on each other. This equilibrium can be explained by the forces of gravity, the walls of the container, and the contact forces between the spheres.
  • #1
skiboka33
59
0
Two identical uniform spheres, each weighing 75 N are at rest on the bottom of a fixed rectangular container. The line of centers of the spheres makes an angle of 35* with the horizontal. Find the forces exerted on the spheres by the container bottom, the container sides as well as the force that each sphere exerts on the other.



ill try to draw a picture for you...



* O O *
* O O O O*
*O O O O*
*O O O O *
* O O *
*************** the circle on the right is above the circle on the left... if you drew a line between the two centers you would have a line 35 degrees to the horizontal...


HELP!
 
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  • #2
wow my picture didnt turn out too well, basically its two circles, the one on the right is above the one on the left ( they don't both fit flat on the bottom of the container )



| |
| --- |
| ( )|
| --- --- |
|( ) |
| --- |
--------------- i hope this one turns out better
 
  • #3
nope, damn, i hope you get the idea
 
  • #4
I take it, then, that the container is too small for the balls to sit next to one another on the bottom but too large to hold them one above the other. Let's assume that the bottom sphere is pressed against the left wall and that the upper sphere is pressed against the right wall.

There are three forces acting on the upper sphere. First, the force of gravity, <0,-75> (vector form: horizont, vertical components). Second, the wall pressing against the sphere: <-a,0> (we don't yet know how hard it is pressing). Third the lower sphere pressing against the upper sphere. That, of course, occurs where the two spheres are tangent and so along the line between their centers: that is 35 degrees to the horizontal. Taking its magnitude to be "b", that will be <b cos(35), b sin(35)> (Be careful about the signs: the force of the lower ball on the upper is up and to the right so both components are positive. Of course, the "cos" and "sin" are from the right triangle formed.) The total force on the upper sphere is <0-a + b cos(35),-75+ 0+ b sin(35)> and, since the upper sphere is not moving, that must be 0: a= b cos(35) and
b sin(35)= 75.
That's surprising: that's enough to calculate a and b right there!
b= 75/sin(35)= 131 Newtons. And then a= 131 cos(35)= 107 Newtons.

Now let's look at the bottom sphere. It has four forces acting on it. First the force of gravity: <0, -75>. Second the force of the wall: <c,0> (it's fairly easy to see that c= a but let's not assume that). Third the force of the upper sphere pressing down on it: <-b cos(35),-b sin(35)>- exactly the force the lower sphere applies to the upper but oppositely directed, of course. Finally, there is the force of the bottom supporting the lower sphere: <0,d>.
The total force on the lower sphere is <c- b cos(35), -75- b sin(36)+ d> and, since the lower sphere is not moving, that must be 0. We must have c= b cos(35) and d= 75+ b sin(35). The "b" we already know: it is 131 Newtons.

We can see that, in fact, that c= b cos(35) is exactly the same as the previous a= b cos(35)- since the only horizontal force is coming from the two sides, those must be the same in equilibrium. We also know that b sin(35)= 75 (the weight of the upper sphere pressing on the bottom one) so c= 75+ b sin(35)= 75+75= 150 Newtons. That's obvious isn't it? The bottom sphere is pressing down on the base with the weight of both spheres.

Making sure we answer all questions: the container bottom is pushing up on the spheres with 150 Newtons, exactly the weight of the two. Each side is pressing in against a sphere with a= c= 107 Newtons and the two spheres are pressing against one another with b= 131 Newtons.
 

1. What is the concept of circles in a box?

The concept of circles in a box refers to the arrangement of multiple circles within a confined space, typically a square or rectangular shape. It is often used in mathematical and geometric problems to explore the relationship between the size and placement of circles within a given space.

2. How do you calculate the area of circles in a box?

To calculate the area of circles in a box, you can use the formula for the area of a circle (A=πr^2) and then multiply it by the number of circles in the box. If the circles are all the same size, you can also use the formula for the area of a rectangle (A=lw) and multiply it by the number of circles.

3. What is the significance of circles in a box in real life?

Circles in a box have various real-life applications, such as in packing and storage, where maximizing the use of space is important. They are also used in the design of computer chips and other electronic components, as well as in the study of optics and light refraction.

4. How can circles in a box help in problem-solving?

Circles in a box can help in problem-solving by providing a visual representation of a problem and allowing for the manipulation and exploration of different arrangements and sizes of circles. This can aid in identifying patterns, making predictions, and finding solutions.

5. Are there any limitations to using circles in a box in problem-solving?

While circles in a box can be a useful tool in problem-solving, they may not always accurately represent real-life situations or provide a complete solution. Additionally, the number and size of circles that can fit in a box may be limited by the size of the box itself, which may not always be the case in practical situations.

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