A sign is hanging from the ceiling on two threads of equal length, which make an angle $\theta$ of 108.5 "degrees" with each other. What is the tension (in N) in each thread, assuming that the force of gravity acting on the sign is 46.5 N?
To find the tension in each thread, we can use the concept of equilibrium. In equilibrium, the sum of all forces acting on an object is zero.
Let's denote the tension in each thread as T. Since the angle between the threads is known, we can split the weight of the sign into two components: one parallel to each thread.
The component of gravity along each thread can be found using the trigonometric relationship:
$\text{Component}_{\parallel} = \text{Force}_\text{gravity} \times \cos(\theta)$
Substituting the given values:
$\text{Component}_{\parallel} = 46.5 \, \text{N} \times \cos(108.5^{\circ})$
To find the tension in each thread, we need to consider that the total vertical force acting on the sign should be balanced by the vertical components of tension:
$2T \times \sin(\theta) = 46.5 \, \text{N} \times \sin(108.5^{\circ})$
Now, we can solve for T by dividing both sides of the equation by 2 and solving for T:
$T = \frac{46.5 \, \text{N} \times \sin(108.5^{\circ})}{2 \times \sin(\theta)}$
Substituting the values:
$T = \frac{46.5 \, \text{N} \times \sin(108.5^{\circ})}{2 \times \sin(108.5^{\circ})}$
Calculating this expression, we find:
$T = 29.1 \, \text{N}$
Hence, the tension in each thread is 29.1 N.