Determine the tension developed in each cord required for equilibrium of the 20-kg lamp.

#### Solution:

Show me the final answers↓

We will start off by drawing a free body diagram focusing on ring D.

Let us write an equation of equilibrium for the y-axis forces.

F_{DE}\text{sin}\,(30^0)\,-\,196.2\,=\,0

(Solve for F_{DE})

F_{DE}\,=\,392.4 N

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

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

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

392.4\text{cos}\,(30^0)\,-\,F_{DC}\,=\,0(solve for F_{DC})

F_{DC}\,=\,339.8 N

We can now draw a free body diagram focusing on ring C.

Again, we will write our equations of equilibrium.

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

(Remember that we already found F_{DC}\,=\,339.8 N)

+\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}

(Simplify)

F_{CA}\,=\,0.88F_{CB} (eq.3)

Substitute this value into eq.1.

(solve for F_{CB})

F_{CB}\,=\,275.1 N

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

F_{CA}\,=\,242.1 N

#### Final Answers:

F_{DC}\,=\,339.8 N

F_{CB}\,=\,275.1 N

F_{CA}\,=\,242.1 N

where did -196.2 came when equation of equlibruium for the y-axis forces ?

Hi!

The 196.2 N comes from 20 kg multiplied by the acceleration due to gravity, 9.81. Thus, we get 20 kg x 9.81 m/s = 192.6 N. Also, note that this is shown in the first force diagram, with the light blue arrow pointing downwards.

Hope that helps 🙂

HOW ABOUT F,df?

F is just mass times the acceleration due to gravity, that’s also the tension in the cord.

what if we did not have the triangle of 3 and 4 and 5 on AC how would you solve it

Something else must be given, like an angle, or the lengths to figure out the angle. Maybe the question already gives the tension in AC? I am not sure unless I see the question. Please see: https://www.youtube.com/watch?v=X9g4G1eBHCA

sorry can u explain how we got From

CB=275.1 N

So you’re just solving for a single variable. If it’s easier, convert the cosine values into decimals, so cos(45) = 0.707. That should be easier to visualize.