Final answer:
The tension force on a 10-meter long rope, weighing 8 kg, needed to achieve a standing wave speed of 4.5 m/s is 16.2 Newtons, calculated using the formula T = v^2 x μ.
Step-by-step explanation:
To determine the tension force on the rope that is necessary to achieve a standing wave with a speed of 4.5 m/s, we can use the wave speed formula that relates the wave speed (v), the linear mass density (μ), and the tension in the string (T). The formula for the wave speed v is given by:
v = √(T/μ)
We first need to calculate the linear mass density of the rope which is the mass per unit length (kg/m). The linear mass density (μ) is calculated by dividing the total mass of the rope by its length:
μ = mass / length = 8 kg / 10 m = 0.8 kg/m
Next, we rearrange the wave speed formula to solve for the tension T:
T = v^{2} × μ
By substituting the given values into the equation:
T = (4.5 m/s)^{2} × 0.8 kg/m = 20.25 N×m/s^{2} × 0.8 kg/m = 16.2 N
Therefore, the tension force on the rope must be 16.2 Newtons to achieve a standing wave speed of 4.5 m/s.