Question

a school is setting up for the science departments annual night of phantastic physics phun. one of the most popular demonstrations is the traveling wave pulse, in which participants pluck one end of a long, taut wire to send transverse wave pulses racing across the room. the goal is for the pulses to travel across the 20.5 m wire in a time of about 0.500 s. the mass of a 1 m long piece of the wire is known to be 0.445 kg. what should the tension t in the wire be in order to achieve the desired wave speed? t = n

Explanation:

Step1: Calculate the wave speed

The wave speed \(v\) is given by the formula \(v=\frac{d}{t}\), where \(d = 20.5\space m\) (distance) and \(t=0.500\space s\) (time).
\(v=\frac{20.5}{0.500}=41\space m/s\)

Step2: Calculate the linear mass density \(\mu\)

The linear mass density \(\mu\) is the mass per unit length. Given that the mass of a \(1 - m\) long piece of wire is \(m = 0.445\space kg\), so \(\mu=\frac{m}{L}\), and since \(L = 1\space m\), \(\mu=0.445\space kg/m\)

Step3: Use the wave - speed formula for a taut string

The formula for the speed of a transverse wave on a taut string is \(v=\sqrt{\frac{T}{\mu}}\), where \(T\) is the tension and \(\mu\) is the linear mass density. Rearranging for \(T\) gives \(T = v^{2}\mu\)
Substitute \(v = 41\space m/s\) and \(\mu=0.445\space kg/m\) into the formula:
\(T=(41)^{2}\times0.445\)
\(T = 1681\times0.445\)
\(T=748.045\space N\)

Answer:

\(748\space N\)