# Conservative forces on a sliding block

1. Feb 13, 2009

### peaceandlove

1. The problem statement, all variables and given/known data
A 4.5 kg block slides along a track from one level to a higher level after passing through an intermediate valley. The track is frictionless until the block reaches the higher level. There a frictional force stops the block in a distance d. The block's initial speed is v0 = 5.3 m/s, the height difference is h = 1.0 m, and μk = 0.613. Find d.

2. Relevant equations
E-mec 1 (object moving) = K1 + U1
E-mec 2 (object stopped) = K2 + U2 + energy lost to friction

3. The attempt at a solution
K1 + U1 - energy lost to friction = K2 + U2
I said that K1=(1/2)mv^2 and U1=mgy and the energy lost to friction is (fk)d; however, I don't know how to get the right side of the equation. I tried using the same formulas and I came up with 1.6313, but apparently that is incorrect.

2. Feb 13, 2009

### Hootenanny

Staff Emeritus
This looks good. So you explicitly we have:

$$\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2 + R\mu_k d$$

And you want to find the distance, i.e. you want to solve this equation for d. What's the first thing you usually do when solving an equation?

3. Feb 13, 2009

### peaceandlove

The first thing I would do is plug in all the values I know.

m=4.5kg
v1=5.3m/s
g=9.8m/s^2
v2=(I would think it to still be 5.3m/s, but I could be wrong)
h2=0 (I think...)
R=(no clue.)
Uk=0.613

4. Feb 13, 2009

### Hootenanny

Staff Emeritus
Good.
Don't worry about these, we'll get to these later.
Hint: You are looking for the distance d when the object has stopped.
What is the expression for the maximum frictional force on an object?
Good.

Personally, I wouldn't plug in any of the values in just yet. Since we're solving for d I would try to make d the subject of the expression.

5. Feb 13, 2009

### peaceandlove

So would v2 be 0 and R=(Us)(Fn)?

6. Feb 13, 2009

### peaceandlove

And d=((1/2)m(v1)^2+mg(h1)-(1/2)m(v2)^2-mg(h2))/R(Uk).

7. Feb 13, 2009

### Hootenanny

Staff Emeritus
Sounds good to me
Yup. However, you can write it in a somewhat nicer fashion:

\begin{align*} d & = \frac{1}{R\mu_k}\left\{\frac{1}{2}mv_1^2 + mgh_1 - \frac{1}{2}mv_2^2 -mgh_2\right\} \\ & = \frac{1}{R\mu_k}\left\{\frac{1}{2}m\left(v_1^2 - v_2^2\right) + mg\left(h_1-h_2\right)\right\} \\ & = \frac{1}{R\mu_k}\left\{\frac{1}{2}m\left(v_1^2 - v_2^2\right) + mg\Delta h\right\}\end{align*}

Does that help?

8. Feb 13, 2009

### peaceandlove

One last thing, what values do we plug in for (Us) and (Fn)? I got (Fn) to be 44.1 (although I could be wrong), but I don't know how to find (Us).

9. Feb 13, 2009

### Hootenanny

Staff Emeritus
Sorry, I misread your previous post. R is simply Fn= 44.1 N.

As a side note, take care with the sign of $\Delta h$, recall it's definition $\Delta h = h_1-h_2$.

10. Feb 13, 2009

### peaceandlove

Oh, and you never got back to clarifying what h1 and h2 are.

11. Feb 13, 2009

### Hootenanny

Staff Emeritus
What do h1 and h2 represent?

12. Feb 13, 2009

### peaceandlove

The change in y, right? So h1 is 1 and h2 is 0?

13. Feb 13, 2009

### Hootenanny

Staff Emeritus
Almost. h1 and h2 represent the initial and final heights respectively. Hence, $\Delta h = h_1-h_2$ represent the change in height. However, you should note that the question states that the final height is greater than the initial height.

14. Feb 13, 2009

### peaceandlove

Oh, so it's the other way around? Meaning (delta)h=-1?

15. Feb 13, 2009

### Hootenanny

Staff Emeritus
Yes.

16. Feb 13, 2009

### peaceandlove

Thank you so much for your help!

17. Feb 13, 2009

### Hootenanny

Staff Emeritus
A pleasure