Revision as of 20:56, 20 August 2014

The problem

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

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

Step 5: Used Newton's second law

in the [math]\hat i[/math] direction

Forest_UCM_NLM#Block_on_incline_with_friction

@@ Line 34: / Line 34: @@
 :<math>\sum F_{ext} = mg \sin \theta - \mu mg= ma_x = m \frac{dv_x}{dt}</math>
-: <math>\int_0^t {g\sin \theta - \mu g} dt = \int_0^v dv <math>
+: <math>\int_0^t {g\sin \theta - \mu g} dt = \int_0^v dv </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} \left ( \sqrt{\frac{-k}{g \sin \theta}} \; v \right )</math>
-:: <math>i \equiv \sqrt{-1}</math>
-:<math>\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 )</math>
-::<math>i\tan^{-1}(icx) = -\tanh^{-1}(cx) = -\tanh^{-1}\left (\frac{\left | b \right |}{a} x\right )</math>
-Identities
-::<math>\tan^{-1}(z) = \frac{i}{2} \log \left ( \frac{i + z}{i-z}\right )</math>
-::<math>\tanh^{-1}(z) = \frac{1}{2} \log \left ( \frac{1 + z}{1-z}\right )</math>
-:<math>\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)</math>
-:<math>t = \frac{1}{\sqrt{gk\sin \theta}} \tanh^{-1} \left ( \sqrt{\frac{k}{g \sin \theta}} \; \;v \right )</math>
-Solving for <math>v</math>
-: v = \tan \left ( \sqrt{gk\sin \theta} i t \right )
-::=
-[[Forest_UCM_NLM#Block_on_incline_with_friction]]
 [[Forest_UCM_NLM#Block_on_incline_with_friction]]

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