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Find an expression for T_1, the tension in cable 1, that does not depend on T_2?

A chandelier with mass m is attached to the ceiling of a large concert hall by two cables. Because the ceiling is covered with intricate architectural decorations (not indicated in the figure, which uses a humbler depiction), the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. Instead, they attached the cables to the ceiling near the walls. Cable 1 has tension T_1 and makes an angle of

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For cable 1, let the tension be T₁

breaking the components of T₁

-T₁cosθ₁---> Horizontal component

T₁sinθ₁----> Vertical component

similarly for T₂

T₂cosθ₂---> Horizontal component

T₂sinθ₂---> Vertical component

Since the chandelier isn't going anywhere, its components should be balanced

for horizontal,

-T₁cosθ₁+ T₂cosθ₂= 0

T₁cosθ₁= T₂cosθ₂ ....(1)

T₁= T₂cosθ₂/ cosθ₁

for vertical,

T₁sinθ₁+ T₂sinθ₂= mg ...(2)

Repace T₁from (1)

T₂{cosθ₂(sinθ₁)/ cosθ₁} + (T₂sinθ₂) = mg

T₂= mg cosθ₁/ {(sinθ₁x cosθ₂) + (sinθ₂x cosθ₁)}

T₂= mg cosθ₁/ sin (θ₁+ θ₂)

Similarly,

T₁= mg cosθ₂/ sin (θ₁+ θ₂)

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Tension In Cable

or

T1 = mgcos(θv2)sin(θv1+θv2)

or

mgcos(θ2)/sin(θ1+θ2)

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You want to use Newton's second law in the x and y direction and set each equation equal to zero since the chandelier isn't accelerating.

FORCES IN THE X:

-T1*cos(θ1) + T2*cos(θ2) = 0

If you look at the figure, you'll notice that the x components of the tensions point in opposite direction.

T1*cos(θ1) = T2*cos(θ2)

Since you want to find an expression for T1 in terms of T2, solve for T2.

==(1)==> T2 = T1*cos(θ1) / cos(θ2)

FORCES IN THE Y:

T1*sin(θ1) + T2*sin(θ2) - mg = 0

==(2)==> T1*sin(θ1) + T2*sin(θ2) = mg

Since you want to find an expression for T1 in terms of T2, substitute equation (1) into equation (2) for T2:

T1*sin(θ1) + T1*cos(θ1)*sin(θ2)/cos(θ2) = mg

But sin(θ2)/cos(θ2) = tan(θ2)

T1*sin(θ1) + T1*cos(θ1)*tan(θ2) = mg

T1[sin(θ1) + cos(θ1)*tan(θ2)] = mg

EXPRESSION FOR T1 INDEPENDENT OF T2:

==> T1 = mg / [sin(θ1) + cos(θ1)*tan(θ2)]

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