A 14.0 kg sign hangs from 2 lengths of rope

[j doe]

Homework Statement

A 14.0 kg sign hangs from 2 lengths of rope, each of which is 70.0 cm and at a 20.0 degree angle from the ceiling.

1) What is the tension of the rope?
2) How could you reduce the tension on the ropes for the same sign without adding another rope?

Homework Equations

The Attempt at a Solution

1) 14.0 kg x 9.817 m/s² = 137.438 N
137.438 N / 2 = 68.719 N
sin20.0 = 68.719 N / x
x = 200.921 N

2) you can reduce the tension by making the ropes longer. this will result in heavier ropes, allowing for weight distribution to be more equal, causing the ropes to have less tension.

are these correct?

 

[Let'sthink]

The first part is correct. For the second part increasing the rope length to increase the angle above 20 degree us equivalent to using another rope. So better we lower the sign by hinging the ropes closer to each other there by reducing the distance between the hinges on the ceiling. Thus when the ropes become vertical and teh distance between them equal to the length of the sign the tension becomes the minimum possible. If we bring the hinges at ceiling nearer the tension will again increase.

 

[j doe]

The first part is correct. For the second part increasing the rope length to increase the angle above 20 degree us equivalent to using another rope. So better we lower the sign by hinging the ropes closer to each other there by reducing the distance between the hinges on the ceiling. Thus when the ropes become vertical and teh distance between them equal to the length of the sign the tension becomes the minimum possible. If we bring the hinges at ceiling nearer the tension will again increase.

so by hinging the ropes closer to each other, does that mean the angle would change? can i say hang the sign at a degree lower than 20 from the ceiling?

 

[Let'sthink]

If you hing them closer angle with celing of each one would increase till it becomes 90 degree when ropes become vertical.

 

[drvrm]

sin20.0 = 68.719 N / x
x = 200.921 N

you wish to reduce the tension so the angle can be made larger say moving towards its(sin theta) maximum then you can decrease the tension

 

[Let'sthink]

Yes. The angle made by each rope with ceiling can be viewed in two ways one will be acute and another obtuse. They become same right angle when the ropes become vertical. The internal angles with ceiling are acute if the distance between hinges is more than the length of sign and obtuse when that distance is less than the length of the sign. So increasing or decreasing the angle has to be seen in this light. For completeness such problems need to be appended with appropriate figure.