A ski gondola is connected to the top of a hill by a steel cable of length 680 m and diameter 1.7 cm. As the gondola comes to the end of its run, it bumps into the terminal and sends a wave pulse along the cable. It is observed that it took 15 s for the pulse to return. What is the speed of the pulse? AND What is the tension in the cable? I thought part 1 was t=distance/speed but the answer was not correct. Please help!
Part one is asking for the speed of the pulse. The equation for speed is v = d/t (or v = s/t depending on your textbook. Velocity = Distance/Time. Since the pulse is traveling from the terminal to the other side, and back to origin, you must take double the distance of the rope. So you are looking at the equation: v = 1360m/15s (since 15s is how long it takes for the pulse to travel from and to the origin.) So the answer to the first part is 90.66 m/s. Part II im still working on, sorry.
To find the speed of the pulse, we can use the formula:
Speed = Distance / Time
In this case, the distance is the length of the cable, which is given as 680 m, and the time is the time it took for the pulse to return, which is 15 s.
So the speed of the pulse is:
Speed = 680 m / 15 s
= 45.33 m/s
Therefore, the speed of the pulse is approximately 45.33 m/s.
Now, let's move on to finding the tension in the cable.
The tension in the cable can be determined using the wave speed and the properties of the cable. The wave speed is the same as the velocity of the pulse, which we have just found to be 45.33 m/s.
The formula to find the wave speed on a stretched string is:
Wave Speed = √(Tension / (linear density))
Where:
- Tension is the force applied to the cable.
- Linear density is the mass per unit length of the cable, which is given by the formula: Linear Density = (π * (Diameter/2)^2) / Length, where Diameter is in meters, and Length is in meters.
Rearranging the formula to solve for the tension:
Tension = Wave Speed^2 * Linear Density
First, we need to find the linear density:
Linear Density = (π * (Diameter/2)^2) / Length
= (π * (0.017 m/2)^2) / 680 m
= 6.0039 * 10^-7 kg/m
Using this value, along with the wave speed (45.33 m/s) we just calculated, we can find the tension in the cable:
Tension = (45.33 m/s)^2 * (6.0039 * 10^-7 kg/m)
= 1.366 * 10^3 N
Therefore, the tension in the cable is approximately 1.366 * 10^3 N.