# Forest UCM NLM BlockOnInclineWfriction

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# 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

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 5: Use Newton's second law

## Motion in the direction described by Newton's second law is:

where is the coefficent of STATIC friction

The 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 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 direction described by Newton's second law is:

if there is no acceleration then

If the object is not moving then

The largest value for the frictional force of an object with no velocity is

The condition that must occur in order for an object with no initial velocity to begin moving with a non-zero accelleration is

The critical angle of incline for the object to start moving becomes

After the object exceeds the above condition is will have a non-zero velocity AND the coefficient of friction will decrease from to

## Newton's second law under the condition that the object has a velocity becomes

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

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