Forest UCM NLM BlockOnInclineWfriction

The problem

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

[math]\displaystyle{ F_f = \mu mg }[/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} = - \mu mg \hat{i} }[/math]

Step 5: Use Newton's second law

Motion in the [math]\displaystyle{ \hat i }[/math] direction governed by Netwon's second law is:

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

[math]\displaystyle{ \int_0^t g \left ( \sin \theta - \mu \right ) dt = \int_{v_i}^{v} dv }[/math]

[math]\displaystyle{ v= v_i - g \left ( \mu -\sin \theta \right ) t }[/math]

The amount of time that lapses until the blocks final velocity is zero

[math]\displaystyle{ t= \frac{v_i}{\left ( \mu - \sin \theta \right ) } }[/math]

After the above time the blocks speed is zero. The friction will change from being kinetic to static after the above time interval.

[math]\displaystyle{ v(t) =\left \{ {v_i - g \left ( \mu -\sin \theta \right ) t \;\;\;\;\;\;\;\; t< \frac{v_i}{\left ( \mu - \sin \theta \right ) } \atop 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t>= \frac{v_i}{\left ( \mu - \sin \theta \right ) }} \right . }[/math]

Forest_UCM_NLM#Block_on_incline_with_friction