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 T1 and makes an angle of θ1 with the ceiling. Cable 2 has tension T2 and makes an angle of θ2 with the ceiling.
Find an expression for T1, the tension in cable 1, that does not depend on T2.
Express your answer in terms of some or all of the variables m, θ1, and θ2, as well as the magnitude of the acceleration due to gravity g. You must use parentheses around θ1 and θ2, when they are used as arguments to any trigonometric functions in your answer.
I understand the formula to plug all of the information into but I am having trouble canceling out T2 and Ive been working on this problem for two days now any help appreciated!
A store sign of mass 4.57 kg is hung by two wires that each make an angle of θ = 40.7° with the ceiling.
Well, sounds like this problem is really hanging over your head! Let me try to lighten the mood and help you out.
To find an expression for T1 that does not depend on T2, we can start by analyzing the forces acting on the chandelier. There are two vertical forces: the tension in cable 1, T1, and the weight of the chandelier, mg (where g is the acceleration due to gravity).
First, let's break down the force of T1 into its vertical and horizontal components. The vertical component of T1 will balance the weight of the chandelier, so we have:
T1 * cos(θ1) = mg (Equation 1)
Now, let's look at the horizontal forces. The horizontal component of T1 is balanced by the horizontal component of T2. Since these two components must cancel each other out, we have:
T1 * sin(θ1) = T2 * sin(θ2) (Equation 2)
Now, it's time to get rid of T2. We can isolate it in Equation 2:
T2 = (T1 * sin(θ1)) / sin(θ2)
Now, substitute this expression for T2 into Equation 1:
T1 * cos(θ1) = mg
Now, divide both sides of the equation by cos(θ1):
T1 = (mg) / cos(θ1)
And there you have it! An expression for T1 that does not depend on T2:
T1 = (mg) / cos(θ1)
So, no need to hang onto your worries anymore. I hope this answer makes you feel buoyant! Keep up the good work!
To find an expression for T1 that does not depend on T2, we can use the fact that the sum of the vertical components of the tension forces must equal the weight of the chandelier.
Let's break down the forces acting on the chandelier:
1) Weight force (mg): This force acts vertically downward and has a magnitude of mg.
2) Tension force from Cable 1 (T1): This force acts upward and makes an angle of θ1 with the ceiling. Its vertical component is T1cos(θ1).
3) Tension force from Cable 2 (T2): This force also acts upward and makes an angle of θ2 with the ceiling. Its vertical component is T2cos(θ2).
Since the chandelier is in equilibrium (not accelerating vertically), the sum of the vertical components of the forces must be zero:
mg + T1cos(θ1) + T2cos(θ2) = 0
Now, we want to express T1 in terms of the given variables only, so we need to eliminate T2 from the equation. We can do this by isolating T2:
T2cos(θ2) = -mg - T1cos(θ1)
Divide both sides by cos(θ2):
T2 = (-mg - T1cos(θ1)) / cos(θ2)
Now, substitute this expression for T2 back into the equation and solve for T1:
mg + T1cos(θ1) + (-mg - T1cos(θ1)) / cos(θ2) = 0
Multiply through by cos(θ2) to get rid of the denominator:
mgcos(θ2) + T1cos(θ1)cos(θ2) - mg - T1cos(θ1) = 0
Now, simplify and collect terms with T1:
- mg + mgcos(θ2) = T1cos(θ1) - T1cos(θ1)cos(θ2)
Factor out T1 on the right side:
- mg + mgcos(θ2) = T1(cos(θ1) - cos(θ1)cos(θ2))
Finally, factor out cos(θ1) from the right side:
- mg + mgcos(θ2) = T1cos(θ1)(1 - cos(θ2))
Divide both sides by (1 - cos(θ2)):
(- mg + mgcos(θ2)) / (1 - cos(θ2)) = T1cos(θ1)
So, the expression for T1, which does not depend on T2, is:
T1 = (- mg + mgcos(θ2)) / (1 - cos(θ2))cos(θ1)
Note: This expression assumes that the cables and the chandelier are massless, and there is no friction or other forces involved.
To find an expression for T1, the tension in cable 1 that does not depend on T2, we can start by considering the forces acting on the chandelier. There are three forces acting on the chandelier: the tension in cable 1 (T1), the tension in cable 2 (T2), and the force due to gravity (mg, where m is the mass of the chandelier).
Let's break down these forces into their components. We'll consider the x-component and y-component separately.
For cable 1:
The x-component of T1 is T1 * cos(θ1).
The y-component of T1 is T1 * sin(θ1).
For cable 2:
The x-component of T2 is T2 * cos(θ2).
The y-component of T2 is T2 * sin(θ2).
For gravity:
The x-component of the force due to gravity is 0 (as gravity acts vertically downwards).
The y-component of the force due to gravity is -mg.
Now, since the chandelier is in equilibrium, the net force in both the x and y directions must be zero.
In the x-direction: T1 * cos(θ1) + T2 * cos(θ2) = 0 (since the chandelier is not moving horizontally).
In the y-direction: T1 * sin(θ1) + T2 * sin(θ2) - mg = 0 (since the chandelier is not moving vertically).
Now, let's solve these equations to eliminate T2.
From the x-direction equation, we can rearrange it as:
T2 * cos(θ2) = -T1 * cos(θ1)
Dividing both sides by cos(θ2) and rearranging, we have:
T2 = - (T1 * cos(θ1)) / cos(θ2)
Now, substitute this expression for T2 into the y-direction equation:
T1 * sin(θ1) - (T1 * cos(θ1) * sin(θ2)) / cos(θ2) - mg = 0
Now, let's solve this equation for T1.
First, let's simplify the equation:
T1 * sin(θ1) - (T1 * cos(θ1) * sin(θ2)) / cos(θ2) = mg
Next, let's multiply both sides by cos(θ2) to eliminate the denominator:
T1 * sin(θ1) * cos(θ2) - T1 * cos(θ1) * sin(θ2) = mg * cos(θ2)
Now, let's factor out T1:
T1 * (sin(θ1) * cos(θ2) - cos(θ1) * sin(θ2)) = mg * cos(θ2)
Using the trigonometric identity sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B), we can simplify further:
T1 * sin(θ1 - θ2) = mg * cos(θ2)
Finally, divide both sides by sin(θ1 - θ2) to get the expression for T1:
T1 = (mg * cos(θ2)) / sin(θ1 - θ2)
So, the expression for T1, the tension in cable 1 that does not depend on T2, is:
T1 = (mg * cos(θ2)) / sin(θ1 - θ2)
I hope this explanation helps you understand how to solve the problem and eliminate T2 while finding the expression for T1.