Finding tension given speed of a pulse, length, radius, and time

In summary, to find the speed of the pulse, we used the equation v = d/t and got a speed of 61.05263158 m/s. To find the tension in the cable, we used the equation T = mv^2 and found a tension of approximately 4675708391 N. We also used the equation m = ρV to calculate the mass of the pulse, and the equation V = πr^2h to calculate the volume of the pulse.
  • #1
tjohn101
93
0

Homework Statement


A ski gondola is connected to the top of a hill by a steel cable of length 580 m and diameter 1.5 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 19 s for the pulse to travel the length of the cable and then return.

(a) What is the speed of the pulse?
61.05263158 m/s
(b) What is the tension in the cable?
? N

Homework Equations


I have no idea.

The Attempt at a Solution


I don't even know where to start. I don't even see how you can calculate tension with the information. Any help is appreciated. Thank you!
 
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  • #2


Hi there,

I can help you with this problem. Let's start by breaking down the problem into smaller parts and using some equations to help us solve it.

(a) To find the speed of the pulse, we can use the equation v = d/t, where v is the speed, d is the distance, and t is the time. In this case, the distance is the length of the cable, which is 580 m, and the time is 19 s. Plugging in these values, we get v = 580 m/19 s = 30.52631579 m/s. However, since the pulse travels back and forth along the cable, we need to divide this by 2 to get the speed of the pulse in one direction. So the final speed of the pulse is 30.52631579 m/s / 2 = 61.05263158 m/s.

(b) To find the tension in the cable, we can use the equation T = mv^2, where T is the tension, m is the mass of the pulse, and v is the speed of the pulse. We already know the speed of the pulse from part (a), so now we just need to find the mass of the pulse. To do this, we can use the equation m = ρV, where ρ is the density of steel and V is the volume of the pulse. The density of steel is 7850 kg/m^3 and the volume can be calculated using the formula V = πr^2h, where r is the radius of the cable (which is half of the diameter, so 0.75 cm) and h is the length of the pulse (which is half of the length of the cable, so 290 m). Plugging in these values, we get V = π(0.75 cm)^2(290 m) = 162.5625 m^3. Now we can calculate the mass of the pulse as m = (7850 kg/m^3)(162.5625 m^3) = 1276050 kg. Finally, we can plug this value into the equation T = mv^2 to get T = (1276050 kg)(61.05263158 m/s)^2 = 4675708391 N. So the tension in the cable is approximately 4675708391 N.

I hope this helps! Let me know if you
 
  • #3


I would approach this problem by first understanding the concept of tension and how it relates to the given variables. Tension is the force applied by a string, rope, or cable when it is pulled from both ends. In this case, the cable is being pulled by the weight of the ski gondola and the force of gravity. The tension in the cable is what allows the pulse to travel along its length.

To calculate the tension, we can use the formula T = F/A, where T is the tension, F is the force applied, and A is the cross-sectional area of the cable. In this case, the force applied is the weight of the ski gondola and the force of gravity. We can calculate this force using the formula F = mg, where m is the mass of the gondola and g is the acceleration due to gravity (9.8 m/s^2).

To find the mass of the gondola, we can use the formula m = ρV, where ρ is the density of steel (7.85 g/cm^3) and V is the volume of the cable. We can calculate the volume using the formula V = πr^2h, where r is the radius of the cable and h is the length of the cable.

Now that we have all the necessary variables, we can plug them into the formula T = F/A to find the tension in the cable. Keep in mind that the cross-sectional area of the cable is given by the formula A = πr^2. Once we have the tension, we can use it to calculate the speed of the pulse using the formula v = √(T/μ), where v is the speed of the pulse and μ is the linear density of the cable (ρV).

Using these equations, we can find the speed of the pulse and the tension in the cable. It is important to note that these calculations are based on certain assumptions and may not be entirely accurate in real-world situations. Therefore, it is always important to double-check our calculations and consider any potential sources of error.
 

What is the formula for finding tension given speed of a pulse, length, radius, and time?

The formula for finding tension is T=μv²πr²/l, where T is the tension, μ is the linear density, v is the speed of the pulse, r is the radius, and l is the length.

How does the speed of the pulse affect the tension?

The speed of the pulse directly affects the tension, as seen in the formula T=μv²πr²/l. A higher speed of the pulse will result in a higher tension, while a lower speed will result in a lower tension.

What is the role of length and radius in determining tension?

The length and radius play a crucial role in determining tension as they are both directly involved in the formula T=μv²πr²/l. A longer length or larger radius will result in a higher tension, while a shorter length or smaller radius will result in a lower tension.

Can the tension be calculated without knowing the speed of the pulse?

No, the tension cannot be calculated without knowing the speed of the pulse. The speed of the pulse is a crucial factor in determining tension, as seen in the formula T=μv²πr²/l.

How can the tension be increased in a given system?

The tension can be increased in a given system by either increasing the speed of the pulse, increasing the length and/or radius, or decreasing the linear density. These factors can be manipulated to increase the tension to a desired level.

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