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3.
A traffic light is suspended by two cables. What
should be the angle between the cables (0) so that the
tension in each cable will be equal to the weight of the
traffic signal light (W=T, = T)?

Respuesta :

MathPhys

Answer:

120°

Explanation:

Draw a free body diagram.  There are three forces acting on the traffic light.  Two tension forces acting along the cables, and weight.

The tension forces have an angle θ between them.  That means each tension force forms an angle of θ/2 with respect to the vertical.  So the y component of each tension force is:

Ty = T cos (θ/2)

Sum of the forces in the y direction:

∑F = ma

Ty + Ty − W = 0

2 Ty = W

Substituting:

2 T cos (θ/2) = W

If W = T, then:

2 W cos (θ/2) = W

2 cos (θ/2) = 1

cos (θ/2) = 1/2

θ/2 = 60°

θ = 120°

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oladayoademosu

The angle between the cables Ф so that the tension in each cable will be equal to the weight of the traffic signal light is 120°

Let

  • Ф = the angle between the cables.
  • T = the tension in cable 1,
  • T' = the tension in cable 2 and
  • W = the weight of the traffic light

Now, the angle between the tension and the vertical is α = Ф/2.

Net force on vertical force on traffic light

The net force vertical force on the traffic light is F = Tcosα + T'cosα - W

Since the traffic light is in equilibrium, we have that F = 0

So, F = Tcosα + T'cosα - W

Tcosα + T'cosα - W = 0

Tcosα + T'cosα = W

Now, since the tension in each cable equals the weight of the traffic light, T = T' = W.

So, Tcosα + T'cosα = W

Wcosα + Wcosα = W

2Wcosα = W

cosα = W/2W

cosα = 1/2

α = cos⁻¹(1/2)

α = 60°

Angle between the cables

Since α = Ф/2

Ф = 2α

Ф = 2(60°)

Ф = 120°

So, the angle between the cables Ф so that the tension in each cable will be equal to the weight of the traffic signal light is 120°

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