Physics question regarding tension?

Question

The mass of 6 kgs is supported by a uniform rope which has a mass of 1.0 kg. What is the tension in the rope at its lower, upper and mid-point?

Please explain how to work out as I thought the tension in the lower, upper and mid-point was the same?

Answer 1

The first thing i will start with is the general equation you will need to use to find the forces. I am sure you have seen it before:

Fg = m * ag   (form of F=ma)

Where:

Fg - force of gravity (Newtons)

m  - mass of the object (kg)

ag - acceleration due to gravity (9.8 m/s^2)

The main trick to this is that the rope has a mass itself. Usually we assume that the rope does not weigh very much in comparison to the mass it is holding. In that case you are right, the tension would be the same everywhere in the rope. In this case though the rope has a mass, this means that each little section of rope will have to support the mass, as well as the rest of the rope that is below it. Think about your arm hanging down. You hand has to hold up your fingers, your forearm has to hold up your fingers and your hand, your upper arm has to hold up your forearm, hand and fingers. Just like this the rope has to hold all the pieces of rope that are below it.

So now what you want to do is draw out the problem, and try to understand what is happening with the forces:

   ----

     |

     |

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  [ m ]  

So at the top we have the attachment point and hanging from that we have the mass at the end of the rope. Now look at the lower end of the rope and what it is holding up. You can see that the only weight it has to support is the weight of the mass. For the end of the rope we can write:

Tlow = 6kg * 9.81m/s^2

Tlow = 58.8 N  ( the tension at the bottom of the rope)

Now for the midpoint of the rope, you can imagine cutting it in the middle and trying to support the weight below it. This will include the mass as well as half of the rope. This is a total of:

m = 6kg + 1/2kg

Using our equation again we have:

Tmid = 6.5kg * 9.8m/s^2

Tmid = 63.7N (the tension at the middle of the rope)

Use the same concept for the top of the rope. You can see that you need to support the weight of the mass, as well as the entire weight of the rope. This means that you will have a total mass of 7kg. Therefore:

Ttop = 7kg * 9.81m/s^2

Ttop = 68.6N ( the tension at the top of the rope)

Hopefully this helps you out. Let me know if you have any other questions.

Answer 2

the tension on all yhe length will be same .

its because the rope is uniform