A sign that has a mass of 10.0kg is suspended by two wires, as shown below. (wire 1 is at a 40 degreee angle with the sign, wire 2 is at no angle) Determine the force exerted by each wire in supporting the sign. \in cluse a free-body diagram as part of your response.
To determine the force exerted by each wire in supporting the sign, we can break down the forces acting on the sign and use trigonometry. Let's start by drawing a free-body diagram:
T1
/________\
/ \
/ \
m| |m
\ /
\ /
\_________/
T2
In the free-body diagram, T1 and T2 represent the tensions in wire 1 and wire 2, respectively. m represents the mass of the sign.
Now, let's analyze the forces acting on the sign. We have the weight (mg) acting downward, and two tension forces (T1 and T2) acting upward. Since the sign is in equilibrium, the vertical component of the tension forces must balance the weight of the sign.
The weight can be calculated using the formula: weight = mass × gravitational acceleration (g). In this case, the weight is given as mg = 10.0 kg × 9.8 m/s^2 = 98.0 N.
To find the forces exerted by each wire, we need to determine the vertical components of these forces. Since wire 1 makes an angle of 40 degrees with the vertical, the vertical component of T1 can be calculated as T1v = T1 × sin(40°). The vertical component of T2 is simply T2.
Now, let's set up the equation:
T1v + T2 = weight
Substituting the values into the equation:
T1 × sin(40°) + T2 = 98.0 N
Since we have two variables (T1 and T2) in one equation, we need another equation to solve for both tensions. The sum of the horizontal components of the tension forces must cancel each other out since there is no horizontal acceleration.
We have:
T1h = T1 × cos(40°)
Now, let's set up the equation for the horizontal components:
T1h - T2 = 0
Substituting the values into the equation:
T1 × cos(40°) - T2 = 0
Now we have a system of two equations:
T1 × sin(40°) + T2 = 98.0 N
T1 × cos(40°) - T2 = 0
We can solve this system of equations simultaneously to find the values of T1 and T2.