A bird sits at rest in the middle of a wire. The wire slopes upward at 5* from the horizontal on either side of the bird. If the wire on both sides of the bird is experiencing a tension force of 106 N, then what is the mass of the bird?
force up from left wire = 106 sin 5
force up from right wire also 106 sin 5
total force up = 212 sin 5
= weight down = m g = 9.81 m
m = (212/9.81)sin 5
To find the mass of the bird, we can use the concept of equilibrium. Since the bird is at rest, the net force on it must be zero in both the horizontal and vertical directions.
Let's break down the forces acting on the bird:
1. The weight of the bird pulls it downward with a force equal to its mass multiplied by the acceleration due to gravity, which is approximately 9.8 m/s^2.
2. The tension forces in the wire pull the bird towards the sloping sides. Since the wire slopes upward at 5 degrees from the horizontal on both sides, we can resolve the tension force into its horizontal and vertical components:
- The horizontal component of the tension force will cancel out because the wire is symmetrical. Therefore, it will not affect the equilibrium of the bird.
- The vertical component of the tension force will contribute to counteracting the weight of the bird.
Now, let's calculate the magnitude of the vertical component of the tension force:
Given:
- Tension force on both sides of the bird = 106 N
- Wire slopes upward at 5 degrees on both sides
The vertical component of the tension force can be found using trigonometry:
Vertical component = Tension force * sin(theta)
where:
- Tension force = 106 N
- theta = 5 degrees
Vertical component = 106 N * sin(5 degrees)
Next, we equate this vertical component to the weight of the bird to satisfy equilibrium:
Vertical component = Weight of the bird
Let's denote the mass of the bird as "m". Then, the weight of the bird is given by:
Weight of the bird = m * 9.8 m/s^2
Now, we can set up the equation:
106 N * sin(5 degrees) = m * 9.8 m/s^2
Solving this equation will give us the mass of the bird.