Acceleration of a hanging mass

[chillpenguin]

I would rather ask a more general question then the specific homework question... Basically I am wondering how to calculate the acceleration of a hanging mass that is attached to another mass on a flat surface. Picture a mass on a table. This mass is attached to a string, which is hooked onto a pulley, and then goes off the edge of the table and is attached to a second mass. I originally thought if it was a friction-less surface that the acceleration would just be gravity no matter what. But I think varying the 2 masses would change things. Also I would like to know how to account for friction. What is the acceleration without friction? What is the acceleration with friction? I need some type of equation with variables so I can solve for any mass values and any coefficient of friction (including friction-less).

 

[mic*]

The net force acting on the object on the table is given by;
F = ma
Where;
F is force
m is mass of the object dangling from the pulley
a is gravitational acceleration - 9.81

Accounting for friction, when the object is stationary subtract
F_fr = μ_sF_N
Where;
F_fr is frictional force
μ_s is static coefficient of friction
F_N is the normal force of your object on the table (mass x 9.81)

Once the object is moving/sliding...
F_fr = μ_kF_N
Where;
μ_k is the kinetic coefficient of friction

 

[haruspex]

The net force acting on the object on the table is given by;
F = ma
Where;
F is force
m is mass of the object dangling from the pulley
a is gravitational acceleration - 9.81

This is wrong if things move. If the hanging mass (m₂) is accelerating, the tension is less than m₂g.

chillpenguin, there are two ways to address this. The safest is to consider FBDs of the two masses separately. If the string is inextensible then they will have the same acceleration. If the pulley requires no torque (frictionless and massless) then the tensions will be the same.
A shortcut is to observe that the driving force is m₂g, and the mass driven (all at the same acceleration) is m₁+m₂. Subtract as necessary from m₂g for the friction.