Determine the maximum mass of the lamp

Determine the maximum mass of the lamp that the cord system can support so that no single cord develops a tension exceeding 400 N.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

First, we will draw a free body diagram focusing on ring D.

Let us write our equations of equilibrium. We will start with the y-axis forces.

+\uparrow \sum \text{F}_\text{y}\,=\,0

F_{DE}\text{sin}\,(30^0)\,-\,9.81m\,=\,0

(Isolate for $F_{DE}$ )

F_{DE}\,=\,\dfrac{9.81m}{\text{sin}\,(30^0)}

(simplify)

F_{DE}\,=\,19.62m

Now, write an equation of equilibrium for the x-axis forces.

\rightarrow ^+\sum \text{F}_\text{x}\,=\,0

F_{DE}\text{sin}\,(30^0)\,-\,F_{DC}\,=\,0

(Substitute the value of $F_{DE}$ we found)

19.62m\text{cos}\,(30^0)\,-\,F_{DC}\,=\,0

(Isolate for $F_{DC}$ )

F_{DC}\,=\,16.99m

We can now switch our focus to ring C. Let us draw a free body diagram again.

Again, we will write our equations of equilibrium.

\rightarrow ^+\sum \text{F}_\text{x}\,=\,0

$16.99m\,-\,F_{CA}\dfrac{3}{5}\,-\,F_{CB}\text{cos}\,(45^0)\,=\,0$ (eq.1)

+\uparrow \sum \text{F}_\text{y}\,=\,0

$F_{CA}\dfrac{4}{5}\,-\,F_{CB}\text{sin}\,(45^0)\,=\,0$ (eq.2)

Let us solve for $F_{CA}$ and $F_{CB}$ .

Isolate for

F_{CA}

in eq.2.

F_{CA}\,=\,\dfrac{F_{CB}\text{sin}\,(45^0)}{\dfrac{4}{5}}

(Simplify)

$F_{CA}\,=\,0.884F_{CB}$ (eq.3)

Substitute this value back into eq.1.

16.99m\,-\,0.884F_{CB}\dfrac{3}{5}\,-\,F_{CB}\text{cos}\,(45^0)\,=\,0

(Solve for $F_{CB}$ )

F_{CB}\,=\,13.7m

Substitute this value into eq.3 to figure out $F_{CA}$ .

F_{CA}\,=\,(0.884)(13.7m)

F_{CA}\,=12.1m\,

let us now look at the results we got:

F_{DE}\,=\,19.62m

F_{DC}\,=\,16.99m

F_{CB}\,=\,13.7m

F_{CA}\,=12.1m\,

From these forces, we can see that cord DE experiences the largest force.

Thus, knowing that the maximum tension a cord can handle is 400 N, we can write:

400\,=\,19.62m

(solve for m)

$m\,=\,20.38$ kg

Maximum mass of the lamp = 20.38 kg