Workers attach a 25.0 kg mass to one end of a 20.0 m long crane, suspending the mass in midair. If the cable has a mass of 12.0 kg, determine the speed of transverse wave at the middle and bottom end of the cable.
To determine the speed of transverse waves at the middle and bottom end of the cable, we need to use the formula for wave speed, which is given by:
v = √(Tension / linear density)
where v is the wave speed, Tension is the tension in the cable, and linear density is the mass per unit length of the cable.
First, let's calculate the Tension in the cable:
Tension = force due to the weight of the mass
= mass of the load * gravitational acceleration
= 25.0 kg * 9.8 m/s^2
≈ 245 N
Next, we can calculate the linear density of the cable:
Linear density = total mass of the cable / total length of the cable
The total mass of the cable is the sum of the mass of the load and the mass of the cable itself:
Total mass of the cable = mass of the load + mass of the cable
= 25.0 kg + 12.0 kg
= 37.0 kg
Total length of the cable = length from top to bottom end + length from bottom to middle
= 20.0 m + 20.0 m
= 40.0 m
Linear density = 37.0 kg / 40.0 m
= 0.925 kg/m
Now we can calculate the wave speed at the middle and bottom end of the cable using the formula:
v = √(Tension / linear density)
For the middle end, the wave speed is:
v_middle = √(Tension / linear density)
= √(245 N / 0.925 kg/m)
≈ 22.72 m/s
For the bottom end, the wave speed is the same, as the tension and linear density are the same along the length of the cable:
v_bottom = v_middle
≈ 22.72 m/s
So, the speed of the transverse wave at the middle and bottom end of the cable is approximately 22.72 m/s.
Tension is not uniform on the cable, due to weight being distributed. So speed along the cable is not uniform...
at the bottom: tension=(20*12)*9.8 N
at the middle: tension=(20+6)*(9.8)N
mass per length (m/l)= 12/20=.6kg/meter
speed= sqrt (tension/(m/L))