Forest UCM NLM BlockOnInclineWfriction

The problem

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

$F_f = \mu mg$

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

$\vec{N} = \left | \vec{N} \right | \hat{j}$

$\vec{F_g} = \left | \vec{F_g} \right | \left ( \sin \theta \hat{i} - \cos \theta \hat{j} \right )= mg \left ( \sin \theta \hat{i} - \cos \theta \hat{j} \right )$

$\vec{F_f} = - \mu mg \hat{i}$

Step 5: Use Newton's second law

in the $\hat i$ direction

$\sum F_{ext} = mg \sin \theta - \mu mg= ma_x = m \frac{dv_x}{dt}$

$\int_0^t g \left ( \sin \theta - \mu \right ) dt = \int_{v}^{v_f} dv$

$v-v_f= g \left ( \mu - \sin \theta \right ) t$

The fall time of the block is

$t= \frac{v}{\left ( \mu - \sin \theta \right ) }$

Forest_UCM_NLM#Block_on_incline_with_friction