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

$F_f = kmv^2$

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

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

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

$\vec{F_f} = - kmv^2 \hat{i}$

Step 5: Used Newton's second law

Motion in the $\hat i$ direction described by Newton's second law is:

$\sum F_{ext} = mg \sin \theta -mkv^2 = ma_x = m \frac{dv_x}{dt}$

Notice a terminal velocity $v_t$ exists when $a_x =0$

$mg \sin \theta -mkv^2 = ma_x = 0$

$\Rightarrow v_t^2 = \frac{g \sin \theta}{k}$

$\int_0^t dt = \int_{v_i}^v \frac{dv}{g\sin \theta - kv^2}$

Integral table $\Rightarrow$

$\int \frac{dx}{a^2 + b^2x^2} = \frac{1}{ab} \tan^{-1} \frac{bx}{a}$

$a^2 = g \sin \theta$

$b^2= -k$

$\int \frac{dv}{g\sin \theta - kv^2} = \frac{1}{\sqrt{-gk\sin \theta}} \tan^{-1} \left ( \sqrt{\frac{-k}{g \sin \theta}} \; v \right )$

$i \equiv \sqrt{-1}$

$\int \frac{dv}{g\sin \theta - kv^2} = \frac{1}{\sqrt{gk\sin \theta}} i \tan^{-1} \left ( \sqrt{\frac{k}{g \sin \theta}} \; \;iv \right )$

$i\tan^{-1}(icx) = -\tanh^{-1}(cx) = -\tanh^{-1}\left (\frac{\left | b \right |}{a} x\right )$

Identities

$\tan^{-1}(z) = \frac{i}{2} \log \left ( \frac{i + z}{i-z}\right )$

$\tanh^{-1}(z) = \frac{1}{2} \log \left ( \frac{1 + z}{1-z}\right )$

$\tan^{-1}(ix) = \frac{i}{2} \log \left ( \frac{i + ix}{i-ix}\right )=\frac{i}{2} \log \left ( \frac{1 + 1x}{1-x}\right ) = i\tanh^{-1}(x)$

$t = \frac{1}{\sqrt{gk\sin \theta}} \tanh^{-1} \left ( \sqrt{\frac{k}{g \sin \theta}} \; \;v \right )$

Solving for $v$

v = \tan \left ( \sqrt{gk\sin \theta} i t \right )

=

Forest_UCM_NLM#Block_on_incline_with_friction