Problem from class - Example 2 from Dynamics Note 2
For a diagram see your class notes.....A 3 kg mass is on a 30 degree inclined plane. It is attached to a pulley system with a 10 kg object hanging straight down on the end of the pulley.
a) Find the acceleration of the 10 kg mass assuming the inclined plane is frictionless. b) Fine the acceleration of the 10 kg mass assuming the inclined plane is not frictionless and has a coefficient of kinetic friction of 0.4.
c) Find the tension in the string.

First draw your free body diagrams for each object.

Analyze forces on 10 kg object on pulley:

Let counterclockwise be positive (remember define your direction for the whole system)

Equation 1

a) Says to find acceleration if the plane is frictionless.......

Analyze forces on 3 kg object PARALLEL to plane: Note you don't need to analyze perpendicular to the plane for part a) because the 3 kg object has no net force perpendicular to the plane (ie. it does not lift off the plane). You would only need to analyze perpendicular if you needed to find the Normal Force as in a friction case (part b)).

    (remember since we defined counterclockwise as +ve, down the plane is +ve)

Equation 2

Equation 1 + Equation 2 (elimination to solve):

b) Now it says to find acceleration if the plane has friction.  Note because we know the system will accelerate UP the plane with no friction we now can define the direction of acceleration and hence the direction of the friction force (ie. if the acceleration is UP the plane the force of friction will be down the plane).  It is often necessary to determine what the acceleration will be in the frictionless case first before you can solve the friction case (since you will need to find out what direction to put the frictional force in).

Analyze forces on 3 kg object PERPENDICULAR to plane (in order to find the Normal Force so you can determine the magnitude of the frictional force):

(net force is 0 since it does not lift off the plane)

Analyze forces on 3 kg object PARALLEL to plane:

Equation 3

Equation 1 + Equation 3 (elimination to solve):

c) Find the Tension (just substitute the acceleration back in equation 1 or 3 to find):