One end of a 14 m long wire having a total mass of .8000 kg is fastened to a fixed support in the ceiling, and a 7.50 kg object is hung from the other end. If the wire is struck a transverse blow at one end, how much time does the pulse take to reach the other end? neglect the variation in tension along the length of the wire.
0.390 s
To determine the time it takes for the pulse to travel through the wire, we need to use the formula for the speed of a wave:
v = √(T/μ)
Where:
- v is the speed of the wave
- T is the tension in the wire
- μ is the linear mass density (mass per unit length) of the wire
First, let's calculate the tension in the wire. Since the wire is in equilibrium when the object is hanging from it, the tension is equal to the weight of the object:
T = m * g
Where:
- T is the tension
- m is the mass of the object
- g is the acceleration due to gravity
In this case, the mass (m) of the object is 7.50 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s². Therefore:
T = 7.50 kg * 9.8 m/s² = 73.5 N
Next, we need to calculate the linear mass density (μ) of the wire. Linear mass density is defined as the mass per unit length:
μ = mass/length
In this case, the mass of the wire is given as 0.8000 kg, and the length is 14 m. Therefore:
μ = 0.8000 kg / 14 m = 0.0571 kg/m (rounded to four decimal places)
Now, we can substitute the values of T and μ into the equation for the wave speed:
v = √(73.5 N / 0.0571 kg/m)
Calculating this gives us:
v ≈ 66.55 m/s
Now, to find the time it takes for the pulse to travel the length of the wire (14 m), we can use the formula:
Time = Distance / Speed
Time = 14 m / 66.55 m/s
Calculating this gives us:
Time ≈ 0.2102 seconds (rounded to four decimal places)
Therefore, it takes approximately 0.2102 seconds for the pulse to travel from one end of the wire to the other.