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A 420-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 120-N rod as indicated in the figure below. The left end of the rod is supported by a hinge and the right end is supported by a thin cable making a 30.0° angle with the vertical. (Assume the cable is connected to the very end of the 6.00-m-long rod, and that there are 2.00 m separating the wall from the sign.) 30.0 ICE CREAM SOP (a) Find the (magnitude of the) tension T in the cable. In order to find the tension in the cable, impose the requirements that the net torque and net force must satisfy at equilibrium. N (b) Find the horizontal and vertical components of the force exerted on the left end of the rod by the hinge. (Take up and to the right to be the positive directions. Indicate the direction with the sign of your answer.) horizontal component vertical component

A 420-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 120-N rod as indicated in the figure below. The left end of the rod is supported by a hinge and the right end is supported by a thin cable making a 30.0° angle with the vertical. (Assume the cable is connected to the very end of the 6.00-m-long rod, and that there are 2.00 m separating the wall from the sign.) 30.0 ICE CREAM SOP (a) Find the (magnitude of the) tension T in the cable. In order to find the tension in the cable, impose the requirements that the net torque and net force must satisfy at equilibrium. N (b) Find the horizontal and vertical components of the force exerted on the left end of the rod by the hinge. (Take up and to the right to be the positive directions. Indicate the direction with the sign of your answer.) horizontal component vertical component

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A 420-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 120-N rod as indicated in the figure below. The left end of the rod is supported by a hinge and the right end is supported by a thin cable making a 30.0° angle with the vertical. (Assume the cable is connected to the very end of the 6.00-m-long rod, and that there are 2.00 m separating the wall from the sign.) 30.00 ICE CREAM SHOP (a) Find the (magnitude of the) tension T in the cable. In order to find the tension in the cable, impose the requirements that the net torque and net force must satisfy at equilibrium. N (b) Find the horizontal and vertical components of the force exerted on the left end of the rod by the hinge. (Take up and to the right to be the positive directions. Indicate the direction with the sign of your answer.) horizontal component vertical component N
Transcribed Image Text:A 420-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 120-N rod as indicated in the figure below. The left end of the rod is supported by a hinge and the right end is supported by a thin cable making a 30.0° angle with the vertical. (Assume the cable is connected to the very end of the 6.00-m-long rod, and that there are 2.00 m separating the wall from the sign.) 30.00 ICE CREAM SHOP (a) Find the (magnitude of the) tension T in the cable. In order to find the tension in the cable, impose the requirements that the net torque and net force must satisfy at equilibrium. N (b) Find the horizontal and vertical components of the force exerted on the left end of the rod by the hinge. (Take up and to the right to be the positive directions. Indicate the direction with the sign of your answer.) horizontal component vertical component N
Expert Solution
Step 1

Given information:

The weight of the rectangular sign plate, Wplate=420 N

The width of the rectangular plate, a=4.00 m

The height of the rectangular plate, b=3.00 m

The weight of the rod, Wroad=120 N

The length of the rod, L=6.00 m

The angle made by the cable with the vertical is,  α=30°

Step 2

The free-body diagram of the given problem is shown below

Advanced Physics homework question answer, step 2, image 1

Here,

T is the tension in the cable

Hx is the horizontal component of the force exerted by the hinge

Hy is the vertical component of the force exerted by the hinge

From the geometry of the given figure, we can calculate the value of θ as follows

90°+30°+θ=180°θ=180°-120°θ=60°

Step 3

(a)

As the rod is not rotating, the torque acting on the signboard will be zero.

-Wplate×L-2-Wrod×L2+T×L×sinθ=0T×L×sinθ=Wplate×L-2+Wrod×L2T=Wplate×L-2+Wrod×L2L×sinθ

Here, the torque due to the signboard and rod is taken as negative because they act in a clockwise direction.

Whereas the torque due to the cable is taken as positive as it is acting in the counter-clockwise direction.

Substitute all the known values

T=Wplate×L-2+Wrod×L2L×sinθ=420×6-2+120×626×sin60°=20405.196=392.61 N

This is the tension in the cable.

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