Question

I can solve these two problems, but I can't figure out the weight of each object with a radius R. How do I find the weight, or mass of the rectangle and small circle?

I'm including the entire problem only for your reference.

Each of the following planar objects is placed, as shown in Fig. 2.13, between two frictionless circles of radius R. The mass density per unit area of each object is σ, and the radii to the points of contact make an angle θ with the horizontal. For each case, find the horizontal force that must be applied to the circles to keep them together. For what θ is this force maximum or minimum?

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Answer 1

In the given problems, we need to find out weight of the objects
Now, given is the mass density per unit area = sigma

so for a rectangle of height L in diagram 1
the width of this reactange is 2*(R - Rcos(theta))
hence area of rectangle is 2LR(1 - cos(theta))
mass of rectange is sigma*2LR(1 - cos(theta))
weight can be then written as 2LR*sigma(1 - cos(theta))*g
where g is acceleration due to gravity

Similiarly for case 2, the radius of the smaller circle is r
so from the diagram we can see
(R + r)cos(theta) = R
hence
r = R(sec(theta) - 1)
where R is radius of the bigger circle
so mass of smaller circle is sigma*pi*r^2
weight is sigma*pi*g*R^2(sec(theta) - 1)^2

Also, for the larger disk, radius is R
so, weight is sigma*pi*R^2*g