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 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 theta_1 with the ceiling. Cable 2 has tension T_2 and makes an angle of theta_2 with the ceiling.What is the tension in T_1 as an expression without using T_2?

Question

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 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 theta_1 with the ceiling. Cable 2 has tension T_2 and makes an angle of theta_2 with the ceiling.What is the tension in T_1 as an expression without using T_2?

Answer 1

To find the tension T_1 as an expression without using T_2, we can consider the forces acting on the chandelier.

Let's denote the weight of the chandelier as W.

The weight W can be represented as the vector sum of the tension forces T_1 and T_2 along the cables, plus the vertical component of the weight:

W = T_1 + T_2 + W_vertical

The vertical component of the weight can be determined using the angle theta_1:

W_vertical = W * cos(theta_1)

By substituting this expression into the equation, we get:

W = T_1 + T_2 + W * cos(theta_1)

Now, we can isolate T_1:

T_1 = W - T_2 - W * cos(theta_1)

Hence, the tension T_1 can be expressed without using T_2 as:

T_1 = W - T_2 - W * cos(theta_1)

Answer 2

To find the tension in T_1, we can analyze the forces acting on the chandelier. Since the chandelier is in equilibrium, the sum of the forces in the vertical direction and the sum of the forces in the horizontal direction must both be zero.

Let's break down the forces acting on the chandelier:

1. Force due to T_1: This force can be split into two components - one vertical and one horizontal. The vertical component is T_1 * cos(theta_1) and the horizontal component is T_1 * sin(theta_1).

2. Force due to T_2: This force can also be split into a vertical and a horizontal component. The vertical component is T_2 * cos(theta_2) and the horizontal component is T_2 * sin(theta_2).

3. Weight of the chandelier: This force is acting vertically downward and is equal to m * g, where g is the acceleration due to gravity.

In the vertical direction, the sum of the forces is given by:

T_1 * cos(theta_1) + T_2 * cos(theta_2) = m * g (equation 1)

In the horizontal direction, the sum of the forces is given by:

T_1 * sin(theta_1) + T_2 * sin(theta_2) = 0 (equation 2)

Now, we want to express T_1 in terms of the other variables, without using T_2.

From equation 2, we can solve for T_2 * sin(theta_2):

T_2 * sin(theta_2) = -T_1 * sin(theta_1)

We can substitute this expression in equation 1:

T_1 * cos(theta_1) + (-T_1 * sin(theta_1) * cot(theta_2)) = m * g

Simplifying further, we have:

T_1 * (cos(theta_1) - sin(theta_1) * cot(theta_2)) = m * g

Finally, we can solve for T_1:

T_1 = (m * g) / (cos(theta_1) - sin(theta_1) * cot(theta_2))

So, the tension in T_1 as an expression without using T_2 is (m * g) / (cos(theta_1) - sin(theta_1) * cot(theta_2)).

Answer 3

Write the equation for vertical equilibrium. It will have terms representing weight (mg), T1 sin theta1, and T2 sin theta2.

By using the equation of horizontal equilibrium, you can define T2 in therms of T1 and the two angles. use that fact to eliminate T2 in the first (vertical equilibrium) equation