# Force diagram of a spinning mass tied to a string

1. Mar 11, 2017

### hndalama

1. The problem statement, all variables and given/known data
In an effort to rev up his class, Mr. H does a demonstration with a bucket of water tied to a 1.3-meter long string. The bucket and water have a mass of 1.8 kg. Mr. H whirls the bucket in a vertical circle such that it has a speed of 3.9 m/s at the top of the loop and 6.4 m/s at the bottom of the loop.

Draw a free body diagram for the bucket for each location

2. Relevant equations

3. The attempt at a solution
At the top of the circle the the mass has gravity and tension going downwards, and at the bottom of the circle the mass has gravity downwards and tension upwards. but is there another force acting parallel to the tension and gravity? I think this would be the force that is causing the bucket to move in circular motion but I also think I might be thinking of a centrifugal force which is a fictitious force.

2. Mar 11, 2017

### PeroK

Correct.

Real forces must have a real physical cause. You have gravity and tension here. Where would another physical force come from?

3. Mar 13, 2017

### hndalama

right but at the top of the circle if there is only gravity and tension both acting downwards then why does the mass not drop straight down?

4. Mar 13, 2017

### PeroK

Because it's moving! It accelerates directly down at this point, but it still has its horizontal speed.

5. Mar 13, 2017

### CWatters

The simple view just has gravity and tension. Tension acting radially.

However in this case the angular velocity of the bucket isn't constant. Its faster at the bottom. So there is angular acceleration which means the tension force (probably) can't be radial all the time.

Its not clear which view the problem requires.

6. Mar 13, 2017

### CWatters

PS Its likely they want the simple view but the other view explains how the bucket might be accelerated/decelerated as it rotates.

7. Mar 13, 2017

### PeroK

I think you'll find that gravity explains the angular acceleration!

8. Mar 14, 2017

### CWatters

Not when I did the sums but perhaps I made a mistake...

Energy in rotating mass = 0.5Iω2
Moment of inertia (for point mass on string) I = mr2
ω = v/r

Substituting for I and ω gives the familiar equation..
Energy = 0.5mv2

Energy at the bottom
= 0.5*1.8*6.42 = 36.9J

Energy at the top
= 0.5*1.8*3.92 = 13.7J

Difference = 36.9-13.7 = 23.2J

However the GPE = 1.8*9.81*2.6 = 45.9J

So the rope must be slowing it down on the way down and speeding it up on the way up. However I'm pretty sure this is over thinking the problem.

9. Mar 14, 2017

### PeroK

The difference in energy at the top and bottom is entirely due to the change in GPE. That's how you'd solve the problem.

10. Mar 14, 2017

### hndalama

To find the difference in energy don't we have to account for the kinetic energy as well. so shouldn't it be KE(bottom) - KE(top) - PE(top) . PE at the bottom is 0.
so using CWatters numbers the change in energy is 36.9J -13.7J - 45.9J = -22.7 J

But how does considering the change in energy relate to whether tension is affecting the angular acceleration?

11. Mar 14, 2017

### PeroK

The waters are certainly muddied now. The change in energy of the bucket means the change in its KE. The change in its overall energy is 0, as the change in GPE is equal and opposite to its change in KE.

The assumption in this problem is that the string can sustain no lateral force, so when moving in a circle the tension is perpendicular to the velocity, hence does no work, hence neither adds nor subtracts from the overall energy. To get the object moving, the string would have to be pulled in a direction not perpendicular to its motion (i.e. not moved in a circle). I assume this problem assumes that steady circular motion has been reached.

Check the change in GPE and KE from the numbers given! My guess is they will be equal. If not, then maybe think again.

12. Mar 14, 2017

### PeroK

Okay, whoever set this problem got the numbers wrong, I think. The gain in KE is almost exactly half the loss in PE, so either:

a) they forgot about the factor of $1/2$ for KE
b) they took the height difference to be $1.3m$ rather than $2.6m$

Either way, the fact that one is precisely twice the other is a bit suspicious.

13. Mar 14, 2017

### hndalama

this only applies if the mechanical/overall energy is conserved. the point I and I think @CWatters are making is that it isn't in this problem.
The calculation I made was to show that mechanical energy is lost. rather than the question being wrong can't we conclude that the loss of energy is due to air resistance?

14. Mar 14, 2017

### CWatters

No I think I agree with perok in that they probably got the numbers wrong. Otherwise I think its a horrible problem.

15. Mar 14, 2017

### PeroK

If you assume the numbers in the question are correct, then you are wasting your time, in my opinion.

16. Mar 14, 2017

### kuruman

I think energy conservation is not the issue here. We can safely assume that Mr. H also does an unspecified amount of non-conservative work as he whirls the bucket. One only has to draw two snapshot FBDs at the top and bottom as the problem asks.