Motion of Block Down an Inclined Plane with Friction

[danago]

http://img297.imageshack.us/img297/1518/88977091un0.gif

I managed to derive a solution, however, it doesn't seem to give me the correct answer. Here is what i did:

I first defined my positive x-axis to go down the slope and my positive y-axis to be upward and normal to the slope. The forces acting on the block before it hits the spring are:

$$\begin{array}{l}<br /> \overrightarrow W = \left( {\begin{array}{*{20}c}<br /> {mg\sin \theta } \\<br /> { - mg\cos \theta } \\<br /> \end{array}} \right) \\ <br /> \overrightarrow N = \left( {\begin{array}{*{20}c}<br /> 0 \\<br /> {mg\cos \theta } \\<br /> \end{array}} \right) \\ <br /> \overrightarrow F = - \mu _k \left| {\overrightarrow N } \right|\frac{{\overrightarrow v }}{{\left| {\overrightarrow v } \right|}} = \left( {\begin{array}{*{20}c}<br /> { - \mu _k mg\cos \theta } \\<br /> 0 \\<br /> \end{array}} \right) \\ <br /> \end{array}$$

Where W is the weight, F is the kinetic friction and N is the normal force.

As it moves down the slope and compresses the spring, it will lost gravitational potential energy, gain elastic potential energy, lose kinetic energy and lose energy due to friction.

$$\begin{array}{l}<br /> \Delta V_g = mg\Delta h = - mg(\Delta x + \delta )\sin \theta \\ <br /> \Delta V_e = \frac{1}{2}k\Delta (x^2 ) = \frac{1}{2}k(\Delta x)^2 \\ <br /> \Delta T = - \frac{1}{2}mv^2 \\ <br /> W_{friction} = - \mu _k mg(\delta + \Delta x)\cos \theta \\ <br /> \end{array}$$

Where V_g/e are the gravitational/elastic potential energies, T is the kinetic energy and W_friction is the work done by the friction on the block i.e. energy lost due to friction. Delta x is the compression of the spring, so delta x + delta is the total distance down the slope which the block will move.

Since energy is conserved within the system:

$$W_{friction} = \Delta T + \Delta V_g + \Delta V_e$$

Substituting my expressions for the different energies into this gives me a quadratic equation in delta x, which has the following solution:

$$\begin{array}{l}<br /> \Delta x = \frac{{ - B \pm \sqrt {B^2 - 4AC} }}{{2A}} \\ <br /> \\ <br /> A = 0.5k \\ <br /> B = mg(\mu _k \cos \theta - \sin \theta ) \\ <br /> C = mg\delta (\mu _k \cos \theta - \sin \theta ) - 0.5mv^2 \\ <br /> \end{array}$$

Anyone see if I am doing anything wrong?

Thanks in advance,
Dan.

 

[danago]

Plugging my values into my solution i get -0.256 and 0.286. I would think that id take the latter as my solution since I've treated delta x as a positive quantity through out the derivation of my solution.

 

[alphysicist]

Hi danago,

That looks right to me; the positive solution would be the answer you wanted for this problem.

 

[danago]

Thanks for the reply The working just got a little messy so i thought perhaps i had made an algebraic error, but i guess the solution given to me could always be at fault.