the problem
Consider a block of mass m sliding down the inclined plane shown below with a frictional force that is given by
- [math]F_f = kmv^2[/math]
200 px
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
200 px
Step 4: Define the Force vectors using the above coordinate system
- [math]\vec{N} = \left | \vec{N} \right | \hat{j}[/math]
- [math]\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]\vec{F_f} = - kmv^2 \hat{i}[/math]
Step 5: Used Newton's second law
in the [math]\hat i[/math] direction
- [math]\sum F_{ext} = mg \sin \theta -mkv^2 = ma_x = m \frac{dv_x}{dt}[/math]
- [math]\int dt = \int \frac{dv}{g\sin \theta - kv^2}[/math]
Integral table [math]\Rightarrow[/math]
- [math]\int \frac{dx}{a^2 + b^2x^2} = \frac{1}{ab} \tan^{-1} \frac{bx}{a}[/math]
- [math]a^2 = g \sin \theta[/math]
- [math]b^2= -k[/math]
- [math]\int \frac{dv}{g\sin \theta - kv^2} = \frac{1}{\sqrt{-gk\sin \theta}} \tan^{-1} \frac{-kv}{g \sin \theta}[/math]
Forest_UCM_NLM#Block_on_incline_with_friction
