A long wooden plank, with mass m = 20 kg and length L = 8.0 m, is hinged to the

Question

A long wooden plank, with mass m = 20 kg and length L = 8.0 m, is hinged to the outside wall of a building by a pin joint. A rope attached to the plank at distance s = 5.0 m from the wall hold the plank in a horizontal configuration, as drawn below. The rope makes a 45 degree angle with the wall.

a) What is the tension in the rope?

b) What is the magnitude of the force exerted by the pin joint on the plank?

c) If the maximum tension that the rope can withstand without breaking is Tmax = 1000 N. How far out onto the plank can a person of mass m = 50 kg safely walk?

Explanation / Answer

a) let T be the tension in the rope

length L = 8 m

distance s = 5 m

resolving into components of tesnion

we get along horizntal path,

T sin 45 * s = mgL/2

L/2 = 8/2 = 4 m

T = (20*9.8*4)/(5*sin 45)

T = 221.7 N is the tension in the rope.

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b) using vertical components of force

we get Fv = mg - T sin theta

Fv = 20*9.8 - 221.7*sin 45

Fv = 39.23 N

now for horizontal compoennt

Fh = T cos theta

Fh Fh = 221.7*cos 45

Fh = 156.7 N

Net force F^2 = Fh^2 + Fv^2

Fnet^2 = (156.7^2+39.23^2)

Fnet = 161.5 N

for direction

use tan alpha = Fv/Fh

tan alpha = (39.23/156.7)

alpha = 14.06 deg with horizontal
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c)

net Torque T = F s sin theta = m g L/2 + M g x

1000*sin 45 *5 = 20*9.8*4+50*9.8*x

solving for x,

we get x = (3535.5-784)/490

x = 5.61 m

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