73. A mobile is formed by supporting four metal butterflies of equal mass m from a string of length L. The points of support are evenly spaced a distance apart as shown in Figure P5.73. The string forms an angle 1 with the ceiling at each end point. The center section of string is horizontal. (a) Find the tension in each section of string in terms of 1, m, and g. (b) Find the angle 2, in terms of 1, that the sections of string between the outside butterflies and the inside butterflies form with the horizontal. (c) Show that the distance D between the end points of the string is





Figure P5.73

Oh, a mobile made of metal butterflies! That sounds quite fancy. Now let's get to solving those questions:

(a) To find the tension in each section of the string, we'll have to consider the forces acting on it. Since the butterflies are of equal mass, the vertical components of the tension forces will balance the weight of each butterfly. So we can say that the tension in each section is equal to m * g / cos(1).

(b) Now let's move on to the angle 2. If we consider the forces acting on the outside butterfly at one end, we'll see that its vertical component will be balanced by the tension in the section of string between it and the inside butterfly. Similarly, the horizontal component will be balanced by the tension in the section of string between the inside butterflies. So we can say that tan(2) = sin(1) / cos(1).

(c) Finally, let's calculate the distance D between the end points of the string. We know that the horizontal components of tension forces in each section of the string add up to zero, since the center section is horizontal. So the horizontal component of the tension in each section is equal to m * g * tan(2). We also know that the distance between the outside butterflies is 2a. So we can say that D = 2a + 2 * (m * g * tan(2)) / (cos(1) * g).

Phew! That was a lot of calculations, but we got through it. I hope my clown-ish explanations made it a bit more fun for you!

Unfortunately, I cannot see Figure P5.73, but I can still try to help you with the information given. Let's break down the problem into parts:

(a) Finding the tension in each section of string:
We can assume that the tension in all sections of the string is the same since the system is in equilibrium. Let's denote this tension as T.

To find the tension in each section of string, we can consider the forces acting on one of the metal butterflies. From the information given, we know that the mass of each butterfly is m, and the forces acting on it are its weight mg and the tension T in the string.

The force acting upwards due to tension T can be resolved into two components: one in the vertical direction and one in the horizontal direction. The vertical component balances the weight of the butterfly, while the horizontal component balances the forces from the other butterflies.

Therefore, we can write the following equations for the vertical and horizontal components of the forces:

Vertical component: T * cos(θ1) + T * cos(θ1) = mg

Horizontal component: T * sin(θ1) + T * sin(θ1) + T * sin(θ2) + T * sin(θ2) = 0 (since the horizontal components cancel out)

Simplifying these equations will give you the tension in each section of string in terms of θ1, m, and g.

(b) Finding the angle θ2:
To find the angle θ2, we need to consider the forces acting on the butterflies at the ends. From the information given, we know that the string forms an angle θ1 with the ceiling at each end.

The vertical component of the force acting on the end butterflies must balance their weights. This gives us the equation:
2T * cos(θ1) = 2mg

To find the angle θ2, we can consider the horizontal component of the forces acting on the end butterflies. The horizontal forces from the outside butterflies must balance the horizontal forces from the inside butterflies.

This gives us the equation:
2T * sin(θ1) = 2T * sin(θ2)

From these equations, you can solve for the angle θ2 in terms of θ1.

(c) Finding the distance D between the end points of the string:
Without the figure, it's difficult to determine the exact distance D. However, assuming that the distance between the supports of the outside butterflies is 'a' and the length of the horizontal section of the string is 'b', we can find D.

The total length of the string can be expressed as:
L = 2a + b

Using the definition of cosine, we can determine the value of b in terms of L and θ1:
b = L - 2a * cos(θ1)

By substituting this value of b into the equation for L, we can express D in terms of a, L, and θ1.

Keep in mind that these solutions are based on assumptions as the figure was not provided. It would be helpful to refer to the figure for a more accurate analysis of the problem.

In order to find the tension in each section of the string, we need to analyze the forces acting on the system.

(a) Tension in each section of the string:
Let's consider the leftmost butterfly. There are three forces acting on it: the tension force T1, the force due to the weight of the butterfly mg, and the horizontal component of the tension force T2.

Since the butterfly is in equilibrium, the sum of the vertical forces should be zero:
T1sin(θ1) + T2sin(θ2) - mg = 0 ...(1)

Similarly, the sum of the horizontal forces should be zero:
T1cos(θ1) = T2cos(θ2) ...(2)

Now, let's consider the rightmost butterfly. There are also three forces acting on it: the tension force T3, the force due to the weight of the butterfly mg, and the horizontal component of the tension force T4.

Applying the same reasoning as above, we can write the following equations:
T3sin(θ1) + T4sin(θ2) - mg = 0 ...(3)
T3cos(θ1) = T4cos(θ2) ...(4)

Now, we need to find a relationship between T1, T2, T3, and T4. Looking at Figure P5.73, we can see that the length of the string between the leftmost butterfly and the rightmost butterfly is L. Therefore, we can write:

L = 2a + D

where a is the spacing between the butterflies and D is the distance between the end points of the string.

Since each butterfly is equidistant from the center, we can write:
a = (D / 2) sin(θ1)

Substituting this into the previous equation, we get:
L = D sin(θ1) + D

Simplifying, we find:
D = L / (sin(θ1) + 1) ...(5)

Now, substituting T1 = T3 = T and T2 = T4 = Th (since the butterflies have equal mass), and simplifying equations (1), (2), (3), and (4), we get:
T * sin(θ1) + Th * sin(θ2) = (m * g) / 2 ...(6)
T * cos(θ1) = Th * cos(θ2) ...(7)

Dividing equation (7) by equation (6), we get:
tan(θ1) = 2 * T / (m * g) ...(8)

Solving equation (8) for T, we find:
T = (m * g) / (2 * tan(θ1)) ...(9)

Similarly, we can solve equation (7) for Th, and we get:
Th = (m * g) / (2 * tan(θ2)) ...(10)

Thus, the tension in each section of the string is given by:
T = (m * g) / (2 * tan(θ1))
Th = (m * g) / (2 * tan(θ2))

(b) To find the angle θ2, we can use equation (7):
T * cos(θ1) = Th * cos(θ2)

Rearranging and solving for θ2, we get:
θ2 = arccos((T * cos(θ1)) / Th)

(c) Finally, to show that the distance D between the end points of the string is given by equation (5):
D = L / (sin(θ1) + 1)