Forest UCM NLM BlockOnIncline

the problem

Consider a block of mass m sliding down the inclined plane shown below with a frictional force that is given by

[math]\displaystyle{ F_f = kmv^2 }[/math]

File:TF UCM InclinedPlaneWfriction.png

Find the blocks speed as a function of time.

Step 1: Identify the system

The block is the system with the following external forces, A normal force, a gravitational force, and the force of friction.

Step 2: Choose a suitable coordinate system

A coordinate system with one axis along the direction of motion may make solving the problem easier

Step 3: Draw the Free Body Diagram

File:TF UCM FBD InclinedPlaneWfriction.png

Step 4: Define the Force vectors using the above coordinate system

[math]\displaystyle{ \vec{N} = \left | \vec{N} \right | \hat{j} }[/math]

[math]\displaystyle{ \vec{F_g} = \left | \vec{F_g} \right | \left ( \sin \theta \hat{i} - \cos \theta \hat{j} \right )= mg \left ( \sin \theta \hat{i} - \cos \theta \hat{j} \right ) }[/math]

[math]\displaystyle{ \vec{F_f} = - kmv^2 \hat{i} }[/math]

Step 5: Used Newton's second law

in the [math]\displaystyle{ \hat i }[/math] direction

[math]\displaystyle{ \sum F_{ext} = mg \sin \theta -mkv^2 = ma_x = m \frac{dv_x}{dt} }[/math]

[math]\displaystyle{ \int dt = \int \frac{dv}{g\sin \theta - mkv^2} }[/math]

Integral table [math]\displaystyle{ \Rightarrow }[/math]

[math]\displaystyle{ \int \frac{dx}{a^2 + b^2x^2} = \frac{1}{ab} \tan^{-1} \frac{bx}{a} }[/math]

Forest_UCM_NLM#Block_on_incline_with_friction