A tight-rope walker walks across a 30.0 m long wire tied between two poles. The center of the wire is displaced vertically downward by 1.0 m when he is half-way across. If the tension in both halves of the wire at this point is 3675 N, what is the mass of the tight-rope walker? Neglect the mass of the wire
To solve this problem, we need to consider the forces acting on the tight-rope walker at the halfway point. Since the wire is displaced downward, a net force is exerted on the walker due to gravity.
We can start by finding the weight of the tight-rope walker (force due to gravity), which is given by:
Weight = mass × acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
Next, we need to consider the tension in the wire. At the halfway point, the wire is exerting an upward force on the walker of 3675 N on both halves of the wire. This tension force can be split into two components: one component pulling up and another component pulling downward (due to the displacement).
Let's assume that the entire length of the wire has uniform mass per unit length (µ). Since the displacement of the wire is 1.0 m and the walker is at the midpoint, the weight of the wire pulling the midpoint downward can be expressed as:
Force of wire = µ × displacement
Similarly, the upward force exerted by the wire can be expressed as:
Force of wire = 2 × (µ × distance)
Since these two forces (tensions) must be equal, we have:
µ × displacement = 2 × (µ × distance)
Simplifying the equation:
1 × µ = 2 × µ × (distance / displacement)
Therefore, µ cancels out, and we are left with:
1 = 2 × (distance / displacement)
We know that the distance is half of the length of the wire (15.0 m) and the displacement is given as 1.0 m. Plugging in these values, we have:
1 = 2 × (15.0 m / 1.0 m)
Simplifying:
1 = 2 × 15.0
1 = 30.0
This equation is inconsistent. It means our assumption that the mass per unit length (µ) of the wire is constant is incorrect. However, we can use this inconsistency to solve for µ.
Let's redefine the tension in the wire as T to make it easier to work with.
At the halfway point, we have the following forces acting on the tight-rope walker:
- T (upward force exerted by the wire)
- T (downward force due to displacement)
- Weight of the walker (force due to gravity)
Since the tight-rope walker is not accelerating vertically, the net force in the vertical direction is zero. Therefore, we can set up the following equation:
2T - T - Weight = 0
Simplifying:
T - Weight = 0
T = Weight
Now, we can substitute T = 3675 N and solve for the weight of the tight-rope walker:
Weight = mass × acceleration due to gravity
3675 N = mass × 9.8 m/s^2
Solving for mass:
mass = 3675 N / 9.8 m/s^2
mass ≈ 375.0 kg
Therefore, the mass of the tight-rope walker is approximately 375.0 kg.