Newton's Second Law, with uncertainties

Newton's Second Law states that


                                   F
   F  =  m*a        or      a  =  ---
                                   m

Yesterday, you made measurements of force, mass and acceleration of a cart and a weight under several circumstances:

weight and cart are motionless
weight is falling as cart rolls towards the pulley
weight is rising as cart rolls away from the pulley

In theory, under each of these circumstances, the force of tension in the string should always equal the mass of the weight times its acceleration. Does it? Let's find out. But let's carefully pay attention to the uncertainties in your measurements.

Cart and weight lying on table, unconnected

A good check of the calibration and accuracy of the instruments is to measure the force and acceleration when everything is just lying on a table, motionless and unconnected. The tension force SHOULD be zero, and the acceleration SHOULD be zero. Here are measurements I made under these circumstances:

            T  =   0.010 +/- 0.014 N
            a  =  -0.010 +/- 0.024 m/s^2

We expect


            T   =    0
           
            a   =    0

Question 1: Is the force zero, within the uncertainties? Is the acceleration zero, within the uncertainties?

Cart and weight in place, but held motionless

When the weight is hanging motionless, we should have

   force          x            y
 ---------------------------------
   gravity        0        - m2*g
 
   string         0        + T
 ---------------------------------
   total          0          0
 ---------------------------------

This means that


                T  =  m2 * g

Look at your measurements, or at the ones I made.

Question 2: Write out both sides of the equation, showing explicitly the uncertainties in each quantity. Calculate the total uncertainty on each side of the equation. Is the tension equal to the weight times g, to within the uncertainties?

Cart rolling towards pulley as weight falls

If we release the cart and weight from a resting position, the weight falls as the cart rolls towards the pulley. We expect

   force          x            y
 ---------------------------------
   gravity        0        - m2*g
 
   string         0        + T
 ---------------------------------
   total          0        - m2*a
 ---------------------------------

This means that


                T  =  m2 * (g - a)

Look at your measurements, or at the ones I made.

Question 3: Write out both sides of the equation, showing explicitly the uncertainties in each quantity. Calculate the total uncertainty on each side of the equation. Is the tension equal to the weight times (g - a), to within the uncertainties?

Cart rolling away from pulley as weight rises

If we push the cart away from the pulley and then release it, we see the weight rise for a short time as the cart continues to roll away from the pulley. We expect

   force          x            y
 ---------------------------------
   gravity        0        - m2*g
 
   string         0        + T
 ---------------------------------
   total          0        - m2*a
 ---------------------------------

Note that even though the cart is rising, its upward motion is slowing down; that means that its acceleration must be NEGATIVE. So we have again


                T  =  m2 * (g - a)

Look at your measurements, or at the ones I made.

Question 4: Write out both sides of the equation, showing explicitly the uncertainties in each quantity. Calculate the total uncertainty on each side of the equation. Is the tension equal to the weight times (g - a), to within the uncertainties?

If you finish early ...

You know what the mass of the hanging weight was: 200 grams. But what was the weight of the cart plus force sensor plus accelerometer plus mass bar?

You can figure this out by drawing two free-body diagrams: make one for the weight, and one for the cart. Consider the experiment when the weight was falling down and dragging the cart towards the pulley.

write down an equation relating the tension in the string to the acceleration of the weight (as shown above)
write down an equation relating the tension in the string to the acceleration of the cart
note that the same string is attached to cart and to weight, so the tension force T must have the same magnitude in each equation (but perhaps different signs...)
note that the cart and weight move together, so their accelerations must have the same magnitude (but perhaps different signs...)
solve for the mass of the cart plus its baggage

For more information

My complete set of measurements (four pages: unconnected, static, falling, rising)
My tutorial on uncertainties
Professor Lindberg's tutorial on uncertainties
Some additional problems involving Newton's Second Law for practice.