Net force holds object in equilibrium

[Newlander]

Homework Statement

"A 6.0 kg box is held at rest by two ropes that form 30-degree angles with the vertical. An external force F acts vertically downward on the box. The force exerted by each of the two ropes is denoted by T. . . . The magnitude of force F is 410 N. The magnitude of force T is closest to: . . ."

m = 6.0 kg
a = -9.8 m/s²
v_i = 0 m/s
f_F = 410 N
f_T = ?

A figure is included in the practice example, but I am unable to upload it. Each rope is denoted by T--and this is key to the questions I have about the problem, noted below.

Homework Equations

w = mg

F_net = ma

cos [theta] = adjacent/hypotenuse

The Attempt at a Solution

w = mg
f_W = mg = 6.0 kg * -9.8 m/s² =-58.8 N

F_net = ma
F_{net, negative-y-axis} = -58.8 N + (-410 N) = -469 N

cos [theta] = adjacent/hypotenuse = 469/T
--> T = 469/cos30 = 542 N

Concerns:
First, I'm not sure if I determined the magnitude of T the correct way. I figured I could use the positive y-axis as the equal but opposite force of the net force of f_w and f_F; this would in turn serve as the "adjacent" aspect of the trig function calculation.
Second, the problem refers to "force T" as if it is a singular force--but in the diagram (and explanation), there are two ropes denoted by T.
Third, the answer is provided: it is 271 N--exactly half of what I thought the magnitude of a *single* T is. I'm guessing I may be on the right path but am missing something because of the double nature of the force T--just not sure how to convey it in my work. (This isn't something to turn in--I just want to understand how to correctly do such a problem.)

 

[rl.bhat]

469 N force is balance by the components of 2 Ts in the vertical direction.
So 2*T*cos(30) = 469.

 

[Newlander]

I think I understand what you mean by the balance of components, as the y-axis divides the two T forces, but . . . I'm still not following, exactly. Any other way to frame it? Would you have approached things similarly?

 

[rl.bhat]

The title of the post says that Net force holds object in equilibrium
Since point of action of two tensions and the weight of the box and the force are in equilibrium, the resultant of two tensions must be equal to the total downward force.
The angle between tensions is 60 degrees. Find the resultant of the tensions and equate it to the total force. That gives you the magnitude of the tension.

 

[rwisz]

I would like to clarify a few things about this problem. First, the statement "net force holds object in equilibrium" is not entirely accurate. It is more accurate to say that the object is in equilibrium when the net force acting on it is zero. In this case, the box is being held at rest by the two ropes, which are exerting equal and opposite forces on the box. This means that the net force on the box is zero, and it is in equilibrium.

To solve for the magnitude of force T, we can use the fact that the box is in equilibrium and that the sum of all forces acting on it must be zero. This means that the force F acting downward must be balanced by the upward forces from the two ropes. Since the ropes are making equal angles with the vertical, we can use trigonometry to find the magnitude of force T. The correct equation to use is T = F/sin(theta), where theta is the angle between the ropes and the vertical.

Using this equation, we can find that T = 410 N/sin(30) = 820 N. This is the total force exerted by both ropes. To find the magnitude of force T for each rope, we can divide this by 2, since they are exerting equal and opposite forces. This gives us a magnitude of force T for each rope of 410 N.

In regards to your concerns, it is important to remember that the force T is not a single force, but rather the sum of two forces exerted by each rope. So when we solve for T, we are finding the total force exerted by both ropes combined. Additionally, the answer provided is for the magnitude of force T for each rope, not the total force. This is why it is half of your calculated value. I hope this helps clarify the problem and how to correctly solve it.