Finding Force for Backpacker in Bear Sling Equilibrium

[Lightsout565]

Homework Statement

Two trees are 7.6m apart. Calculate the magnitude of force a backpacker must exert to hold a 16-kg backpack so that the rope sags at its midpoint by (a) 1.5m, (b) 0.15m.

http://2.bp.blogspot.com/_qDvB7jRAneY/SrLIlGBVq1I/AAAAAAAAAVA/J8UdxspkfSk/s1600-h/bear+sling.bmp"

Homework Equations

$$\Sigma$$F_y = 0

The Attempt at a Solution

I tried to figure the angle measure and get a component of the force but I think that's totally wrong. For part a I ended up getting 57.6N

 

[PhanthomJay]

Homework Statement

Two trees are 7.6m apart. Calculate the magnitude of force a backpacker must exert to hold a 16-kg backpack so that the rope sags at its midpoint by (a) 1.5m, (b) 0.15m.

http://2.bp.blogspot.com/_qDvB7jRAneY/SrLIlGBVq1I/AAAAAAAAAVA/J8UdxspkfSk/s1600-h/bear+sling.bmp"

Homework Equations

$$\Sigma$$F_y = 0

The Attempt at a Solution

I tried to figure the angle measure and get a component of the force but I think that's totally wrong. For part a I ended up getting 57.6N

That's not right, please show your work in determining the angle and tension in the wire. Note that the weight of the 16 kg backpack is vertically distributed equally to the rope section on each side of it. The rope tension can then be calculated using simple trig.

 

[Lightsout565]

(a)
(3.8m)² + (1.5m)² = x²
x = 4.085m

F_g = F_tsin$$\theta$$
(16kg)(9.8 m/s²) = F_t(1.5/4.085)
F = 427N

Correct?

 

[PhanthomJay]

(a)
(3.8m)² + (1.5m)² = x²
x = 4.085ml

yes, good

F_g = F_tsin$$\theta$$
(16kg)(9.8 m/s²) = F_t(1.5/4.085)
F = 427N

Correct?

No, the backpack weight splits vertically 1/2Fg to each side. Draw a free body diagram of the rope joint to prove this using Newton 1. Your result is off by a factor of 2.

 

[Lightsout565]

But the questions asks "Calculate the magnitude of force a backpacker must exert", which is the sum of the two tensions forces. So wouldn't my answer be correct?

 

[PhanthomJay]

But the questions asks "Calculate the magnitude of force a backpacker must exert", which is the sum of the two tensions forces. So wouldn't my answer be correct?

Why would you sum them? The rope tension around an ideal pulley is the same on both sides of the pulley. Would you say that the rope tension at the right hand tree is 427 N or 213.5 N? Draw a free body diagram of the rope at the tree connection.