Question regarding tension in a string.

In summary, the tension in a string with a weight hanging from it is not simply equal to the weight of the hanging object multiplied by gravity, as the mass can be accelerating. To find the tension, we can use equations for torque and angular velocity, and also check our work using energy methods. The correct tension in this specific example is -2.575N.
  • #1
p75213
96
0
I have a cylinder wound with string. The free end of the string has a weight dangling from it. Is the tension in the string simply mass of the weight*gravity?
 
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  • #2
Ignoring the weight of the string, then the tension in the free part of the string is equal to the weight (mass x gravity).
 
  • #3
A cylindrical spool with mass 0.5kg and radius 0.5m is wrapped with 4m of string. A 5kg mass is hung from the end of the string. How fast will the spool be rotating when all of the string has been pulled off?

angular velocity^2= (initial angular velocity)^2 +2*(angular acceleration)*theta.
omega^2=initial_omega^2+2*alpha*theta
initial angular velocity=0
theta=8
omega^2=2*alpha*8
Find angular acceleration (alpha).


Let T be the tension in the string.
T-mg=ma
T-5*9.8=5a
T-49=5a
T-49=5r*alpha (tangental accel = r*alpha)
T-49=2.5*alpha (a)
----------------------------------
Torque=I*alpha
Torque=rT
I*alpha=rT
0.0625alpha=0.5T
T=0.125alpha
----------------------------------
Substitute T=0.125alpha in (a)
0.125alpha-49=2.5*alpha
-2.375alpha=49
alpha=-20.63rad/s^2
------------------------------------
omega=sqr(2*20.63*8)
omega =18.17rad/s
------------------------------------
That's the worked solution. Now let's look at T - equation (a)
T-49=2.5*alpha
T=2.5*alpha+49
T=100.6N
I would have thought T was equal to mg=5*9.8=49N
 
  • #4
p75213 said:
I have a cylinder wound with string. The free end of the string has a weight dangling from it. Is the tension in the string simply mass of the weight*gravity?
No, not if the mass is accelerating.
 
  • #5
p75213 said:
That's the worked solution. Now let's look at T - equation (a)
T-49=2.5*alpha
T=2.5*alpha+49
T=100.6N
I would have thought T was equal to mg=5*9.8=49N
Some points:
(1) Redo your solution as you've made an error. You have the tension greater than the weight of the hanging mass. That would mean the mass rises instead of falls!
(2) If the tension equaled mg, then the hanging mass would be in equilibrium.
(3) You can check your work by solving the problem using energy methods.
 
  • #6
I see. That makes sense. That's what I was missing. The tension includes the acceleration of the string.
 
  • #7
Doc Al said:
Some points:
(1) Redo your solution as you've made an error. You have the tension greater than the weight of the hanging mass. That would mean the mass rises instead of falls!
(2) If the tension equaled mg, then the hanging mass would be in equilibrium.
(3) You can check your work by solving the problem using energy methods.

Good points. Yes 100N didn't make sense to me either. (-2.575N)
 

1. What causes tension in a string?

Tension in a string is caused by an external force acting on it, such as pulling or stretching the string.

2. How is tension calculated in a string?

Tension in a string can be calculated using the formula T = F * L, where T is the tension in the string, F is the force applied to the string, and L is the length of the string.

3. Why is tension important in string instruments?

Tension is important in string instruments because it determines the pitch and sound quality of the instrument. The higher the tension, the higher the pitch and the brighter the sound.

4. How does tension affect the vibrations of a string?

Tension affects the frequency of vibrations in a string. A higher tension results in a higher frequency of vibrations, producing a higher pitch.

5. Can tension in a string be too high?

Yes, tension in a string can be too high and can result in the string breaking. It is important to find the appropriate tension for each string in order to produce the desired sound and prevent damage to the string.

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