Another physics question

[pizza1994]

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A 1000 kg gondola is operted on a cable between two towers 340 m apart. When the gondola is exactly between the towers, it is 100 m below their height. What is the tension in the cable at this midpoint?

a) 5 kN
b) 8 kN
c) 10 kN
d) 20 kN

the Answer is C)....can someone explain why?

 

[getfat]

If the gondola isnt moving then only two forces are acting on the gondola; Tension and gravity,
Gravity= (mass) (acceleration) = (1000kg)(10)
Height doesnt matter unless you are trying to find gravitational potentional potential energy
The distance between them could only work if maybe you wanted to know the work required to go from tower to tower,idk that factoid seemed pretty useless.

 

[pizza1994]

ya so I agree that theres tension and gravity. but the princeton book splits tension into two components. apparently the 10,000 N gravity gets split into 5000 and 5000 N because of the two vertical tension componenets. but my problem is that I dont see two vertical tension components. I only see one tension componenet....

 

[el_duderino]

If the gondola isnt moving then only two forces are acting on the gondola; Tension and gravity,
Gravity= (mass) (acceleration) = (1000kg)(10)
Height doesnt matter unless you are trying to find gravitational potentional potential energy
The distance between them could only work if maybe you wanted to know the work required to go from tower to tower,idk that factoid seemed pretty useless.

I don't think this is right, because there would be more tension in the rope if the gondola were hanging 1 cm below the top of the towers than if it were hanging 200m below.

I haven't done this type of problem in a couple of years now, but I drew a free body diagram. The tension in the rope off to the upper left is X. The vertical component of the force is 1000 * 9.8 = 9.8 kN. The horizontal is therefore given in t sin(theta) = 9.8 kN, and theta is about 30 degrees. sin(30) is 0.5, so t ~ 20 kN, divided between left and right components. So that's 10 kN.

 

[el_duderino]

OK I thought about it for a few minutes and here's a more straightforward way to think about it.

Imagine the forces on the gondola. 9.8 kN straight down, then t1 to the upper left and t2 to the upper right. t1 and t2 both have a horizontal and vertical component. Since t1 = t2 (the angle is the same), the vertical component of each is 4.9 kN.

We determine the angle using tan theta = 100/170, so theta is about 30. sin(30) = 4.9 kN / t1. sin(30) is 0.5 so t1 = t2 = 9.8 kN.

The trick is to realize that though you have essentially two forces acting on the gondola: gravity, and the vertical component of the tension. However, the vertical component of the tension is really the sum of t1 and t2, each of which has a horizontal component that cancels out the other.

 

[pizza1994]

Yeah so I agree with what you are saying but my question is why is there a t1 to the upper left and a t2 to the upper right? You would have that if the gondola was attached by two ropes on a diagonal.

That is the part that confuses me.

 

[el_duderino]

Well think about it for a second. If the gondola were simply sitting on a hook in the middle of a continuous cable, would the forces be different versus if there were two individual ropes going to each tower? Of course not. Imagine you're a "particle of rope." As part of the rope, you feel the "tension" in the rope as forces pulling you in opposite directions that cancel each other out. Or imagine the rope as a long line of people holding hands. Each person feels two equal and opposite pulling forces from the people on either side.

 

[pizza1994]

ah okay I think what you are trying to say is that the forces wont be different because you form right triangles?

 

[el_duderino]

Correct. You can split the total force into two separate right triangles, which - when you add them together - give you the total force.

 

[pizza1994]

ah okay now you make sense! Thanks so much for your help!