• Paradigm:  The modified Atwood's machine.  

The modified Atwood's machine is a cart towed across a tabletop by a hanging mass.  We observed the cart's motion, and many students noticed that it visibly accelerated.  Recall from the dry ice block that constant force begets constant acceleration.  You suggested that changing the hanging mass or changing the mass of the cart would affect the acceleration. We performed two experiments to test this. 

In our analysis we ignored the mass of the string, inertia of the pulley, and we tilted the track to minimize or eliminate the effect of friction in the wheel bearings and pulley.  When given a brief push, the cart rolled with a relatively constant velocity, evidence that friction did not have a significant effect

We did two experiments.  In experiment 1, we changed the hanging mass without changing anything else.  This meant that when we took mass off of the hanging mass we put it on the cart, or vice versa.  Since we changed the hanging mass, that changed the force of gravity on the hanging mass, which changed the towing force on the cart.  We plotted acceleration vs. towing force.

The equation for this graph is

Acceleration_system=1/mass_system*Towing Force  OR Newton's 2nd Law

In the experiment 2 of the paradigm lab, we investigated what happens if you change mass of the cart, but keep everything else the same.

The equation for this graph is

Accel_system=Net force/mass of cart

Since we noted that the coefficient was slightly less than the towing force, it made sense to call it net force.  Some friction was present, and the mass/inertia of the string and pulley acted to reduce the value of the coefficient.  In an ideal experiment, it would have been exactly equal to the towing force.

Short Summary: the acceleration of the system is proportional to the gravitational force on the hanging mass and inversely proportional to the system mass (cart and hanging mass).

Random Notes

• Forces are vectors.  They follow the rules of vectors.  See the text and the vectors reading for more info.

• To solve force problems, we follow a time-honored procedure.  First, draw a free-body diagram, depicting all the forces acting on the object being considered.  Usually in this class, most objects will be subject to a gravitational force and to a force from everything that touches them.  Second, write "sum of the forces" equations for the forces.  Be sure to only add x (horizontal)  forces to x forces, and y (vertical) forces to y forces.  Third, solve the equations for the desired quantity(ies).

• Newton's second law: the sum of the forces is always equal to the Net Force is equal to mass_system*acceleration_system.  We are interested in the amount (and direction, too) of  "leftover" force that acts on the system  under consideration.  In Unit 4, there was no leftover force, the sum of the forces was always zero.  If the sum of the forces is zero, the acceleration of the system is zero, and you have either constant or zero velocity.
• If there is a non-zero net force, it is equal to ma (Newton's Second Law), and the system accelerates in the direction of the net force.  A non-zero net force causes acceleration.  By net force we do not mean a "real" force, we always mean the sum of the forces.
• The net acceleration is what the system is observed to do.  If the net force is zero, the system either remains at rest or moves at constant velocity.

18 March 2005