# Homework Help: Block in a verticle circle with friction

1. Feb 12, 2012

### GGDK

1. The problem statement, all variables and given/known data
A block of mass 15g enters the bottom of a circular, vertical track with a radius R = 0.5m at an initial velocity of 4/ms. If the block loses contact with the track at an angle of  = 130, what is Wk, the work done by kinetic friction?
http://img94.imageshack.us/img94/1760/problem1t.png [Broken]

2. Relevant equations
Work Energy theorem: (1/2)m(Vfinal)^2 - (1/2)m(Vinitial)^2
Work = Force * Displacement

3. The attempt at a solution
The 3 forces acting on the block: normal force, gravity, and friction.
http://img43.imageshack.us/img43/442/fbodydia.png [Broken]
Before block loses contact,
Gravity has a y-component of .5(sin(40))
Normal force has a y-component of .5(sin(40)) and an x-component of .5(cos(40))

The initial kinetic energy is (1/2).015(4)^2 which equals .012 Joules.
Not really sure where to progress from here.

Last edited by a moderator: May 5, 2017
2. Feb 12, 2012

### PeterO

Since the block is about to leave the track there will be no Normal reaction force. It is the necessity of the reaction force that ensures the block remains in contact with the track up to that point.

Thus the component of the weight force towards the centre will be exactly the centripetal force requires.

Centripetal force is mv2/R , and since R is 0.5 this gives 2mv2.

This just happens to be 4 times the kinetic energy at the point; so we can calculate the kinetic energy from the component of the weight force.

You can easily calculate the Kinetic energy at the bottom. There will be a certain amount of this transformed to potential energy, as it is higher up, and still the remaining kinetic energy above.
The "missing" energy is work done by kinetic friction.

Last edited by a moderator: May 5, 2017
3. Feb 12, 2012

### GGDK

Thanks! So to double check would this be

mg = 9.8(.015) = 4KE
[9.8(.015)]/4 = (1/2)(0.015)(Vfinal)^2
4.9 = V^2
Vfinal = 2.21359m/s

(1/2)(0.015)(2.21359)^2 - (1/2)(0.015)(4)^2 = -.08325 Joules

4. Feb 12, 2012

### PeterO

First line is wrong: mg is the whole force of gravity, only the component of it towards the centre is used.

read my earlier post again - I said "component of the weight force"

5. Feb 12, 2012

### PeterO

Also you are not interested in the final speed; you only needed the final Kinetic Energy - which you re-calculated on the very next line

6. Feb 12, 2012

### galaticman

If you are doing the same problem that I am doing for PHYS 170 im pretty sure that the radius is 0.3m not 0.5m

Last edited by a moderator: May 5, 2017
7. Feb 12, 2012

### GGDK

So would the component of gravity just be .5*sin(40)?

8. Feb 12, 2012

### GGDK

Yep, I changed the number on so I could try the problem again using different values.

9. Feb 12, 2012

### PeterO

I saw this 0.5 earlier but ignored it as a typo.

Where are you getting 0.5 from?

EDIT: I think I see - you are finding a component of the Radius, and calling it a component of the weight .

10. Feb 12, 2012

### GGDK

Oops didn't even realize that!

[9.8(0.015)sin(40)]/4 = KE
KE = .023622 Joules
Final - Initial = Total KE
.023622 - .12 = -.096378 Joules

11. Feb 12, 2012

### PeterO

Getting close: Don't forget a lot of that energy has not been "lost", merely converted to Potential energy because it is higher.

NOTE: even without friction, the block will fall off at some point if it wasn't going fast enough to begin with. Indeed if it is going too slow it won't even reach the 90 degree position, so just stop then slide back down the track.

12. Feb 12, 2012

### lele44

why did you divide by velocity? (4)

13. Feb 12, 2012

### GGDK

I didn't divide by velocity.

In the example problem centripetal force is (mv^2)/r with r as .5 so it would be equal to 2mv^2 which is basically 4 times kinetic energy because KE is (1/2)mv^2.

14. Feb 12, 2012

### PeterO

Great response GGDK - it shows you understood what I was saying.

Just to let you know my original logic.

The block enters the loop with a certain amount of Kinetic Energy.
As it moves up the loop, some of that energy is converted to Potential energy, some is lost to friction, and some of it remains as kinetic energy.
The PE is easy(ish) to calculate, the trick is to calculate how much Kinetic energy remains.
The first thing I addressed in my first post was calculating that - using that convenient expression you quoted above.

NOTE:
I have always been intrigued by the similarity of the Kinetic Energy and Centripetal Force expressions; (1/2)mv2 (or mv2/2) and mv2/R ; meaning that the numerical value of one can be simply transformed to the numerical value of the other.

This "connection" is useful for analysing a loop-the-loop in a roller coaster.
When a cart creeps over hill, plunges to the ground level then swings up through a loop, the kinetic energy at bottom and top of the loop can be calculated from change in PE. - so just mgΔh.

That expression gives mv2/2 , and the centripetal force resulting at each point is mv2/R - so it is easy to convert to the forces involved, without actually bothering with how fast the roller coaster is travelling at the time.