Forest UCM NLM BlockOnIncline
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]
- \int dt = \int \frac{dv}{g\sin \theta - mkv^2}