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 j }[/math] direction described by Newton's second law is:
- [math]\displaystyle{ \sum F_{ext} = N - mg \cos \theta = ma_y = 0 }[/math]
- [math]\displaystyle{ N = mg \cos \theta }[/math]
- [math]\displaystyle{ F_f \le \mu_s N = \mu_s mg \cos \theta }[/math] where [math]\displaystyle{ \mu_s }[/math] is the coefficent of STATIC friction
The [math]\displaystyle{ \le }[/math] indicates that STATIC friction will be a force that is suficient to keep the block from moving. STATIC friction has a maximum value. If the sum of the other forces exceeds the static friction force, then the object will move, and the coeffiicent of kinetic friction [math]\displaystyle{ (\mu_k) }[/math] will be used to describe the motion.
What is the condition to satisfy for the object to move down the inclined plane?
Motion in the [math]\displaystyle{ \hat i }[/math] direction described by Newton's second law is:
- [math]\displaystyle{ \sum F_{ext} = mg \sin \theta - F_f= ma_x = m \frac{dv_x}{dt} }[/math]
if there is no acceleration then
- [math]\displaystyle{ mg \sin \theta = F_f }[/math]
If the object is not moving then
- [math]\displaystyle{ F_f \le mg \sin \theta }[/math]
The largest value for the frictional force of an object with no velocity is
- [math]\displaystyle{ F_f = \mu_s N = \mu_s mg \cos \theta }[/math]
The condition that must occur in order for an object with no initial velocity to begin moving with a non-zero accelleration is
- [math]\displaystyle{ F_f = \mu_s N = \mu_s mg \cos \theta < mg \sin \theta }[/math]
The critical angle of incline for the object to start moving becomes
- [math]\displaystyle{ \theta_c = \tan^{-1}\left( {\mu_s mg}\right ) }[/math]
After the object exceeds the above condition is will have a non-zero velocity AND the coefficient of friction will decrease from [math]\displaystyle{ \mu_s }[/math] to [math]\displaystyle{ \mu_k }[/math]
- [math]\displaystyle{ F_f = \mu_k N = \mu_k mg \cos \theta }[/math]
Newton's second law under the condition that the object has a velocity becomes
- [math]\displaystyle{ \sum F_{ext} = mg \sin \theta - F_f= ma_x = m \frac{dv_x}{dt} }[/math]
- [math]\displaystyle{ \sum F_{ext} = mg \sin \theta -\mu_k mg \cos \theta= ma_x = m \frac{dv_x}{dt} }[/math]
- [math]\displaystyle{ \int_0^t g \left ( \sin \theta - \mu_k\cos \theta \right ) dt = \int_{v_i}^{v} dv }[/math]
- [math]\displaystyle{ v= v_i - g \left ( \mu_k\cos \theta -\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_k\cos \theta - \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_k\cos \theta -\sin \theta \right ) t \;\;\;\;\;\;\;\; t< \frac{v_i}{\left ( \mu_k\cos \theta - \sin \theta \right ) } \atop 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t>= \frac{v_i}{\left ( \mu_k\cos \theta - \sin \theta \right ) }} \right . }[/math]