- #1
FredericChopin
- 101
- 0
Homework Statement
http://imgur.com/FZM5gqC,RlLeGmP#1
http://imgur.com/FZM5gqC,RlLeGmP#0
Homework Equations
[itex]f^k_{AB} = \mu_k N_{AB}[/itex]
[itex]W_{f,i} = \int_{r_i}^{r_f} \vec{F} \cdot d\vec{r}[/itex]
[itex]E^{mech}_f = E^{mech}_i + W^{NC}_{f,i}[/itex]
[itex]U_{elastic} = \frac{1}{2} k x^2[/itex]
The Attempt at a Solution
Since the blocks are at rest after the release of the spring, the final mechanical energy, [itex]E^{mech}_f[/itex], is 0. The initial mechanical energy, [itex]E^{mech}_i[/itex], is the elastic potential energy of the spring ([itex]U_{elastic} = \frac{1}{2} k x^2[/itex]). There are no external non-conservative forces acting on the box-block-spring system, but there is the internal non-conservative force of kinetic friction acting on the box and block ([itex]W^{NC}_{f,i} = \int_{r_i}^{r_f} \vec{f^k_{AB}} \cdot d\vec{r}[/itex]). The force of kinetic friction acts through a displacement [itex]d[/itex] in the same direction as force, so [itex]W^{NC}_{f,i} = \int_{0}^{d} \vec{f^k_{box,block}} \cdot d\vec{r} = f^k_{box,block}d[/itex].
Let's consider the system of the box, the block, and the spring.
The final mechanical energy of this system will be:
[itex]E^{mech}_f = E^{mech}_i + W^{NC}_{f,i}[/itex]
Substituting in terms:
[itex]0 = U_{elastic} + f^k_{box,block}d[/itex]
, which becomes:
[itex]0 = \frac{1}{2} k x^2 + \mu_k N_{box,block}d[/itex]
The block is not accelerating in the vertical direction, and so due to Newton's Second Law, [itex]N_{box,block}[/itex] must be equal in magnitude to [itex]m_{box}g[/itex]:
[itex]0 = \frac{1}{2} k x^2 + \mu_k m_{box}gd[/itex]
Solving for [itex]\mu_k[/itex] yields:
[itex]\mu_k = \frac{-kx^2}{2m_{box}gd}[/itex]
It's strange that there is a negative sign in the answer as [itex]\mu_k[/itex] should be a positive scalar. It also turns out this answer is incorrect.
What went wrong?
Thank you.