Force problems, friction, and uncertainties in slopes

Let's start with some problems involving forces. For each one, draw a free-body diagram and make a table showing all the forces in two directions.

Slippery box A, with mass mA, and slippery box B, with mass mB, lie on a floor and slide without friction. You push them horizontally with a force P to the right.
1. What is the acceleration of the boxes?
2. What is the force of box A on box B?
3. What is the force of box B on box A?
4. Given mA = 40 kg, mB = 10 kg, and P = 100 N, what are the numerical values to each answer?
A slippery box of mass M=10 kg sits on a ramp of angle theta=20 degrees.
1. Draw a set of coordinate axes which are tilted to match the ramp, so that the x-axis goes parallel to the slope and the y-axis goes perpendicular to it.
2. What is the net force on the box in each of these tilted directions?
3. What is the acceleration of the box down the ramp?
4. The box is held at the top of a ramp and released. It slides down to the bottom in 24 seconds. If the sloped surface of the ramp is 12 meters long, what is its tilt?
A cart of mass m1 lies on a slipperly tabletop. The cart is connected by a string to a hanging weight of mass m2. The cart is held motionless, then released. It rolls without friction as the weight falls.
1. What is the acceleration of the weight downwards?
2. What is the acceleration of the cart to the right?
3. What is the tension in the string?

Friction

Some notes on how physicists define friction:

Introduction to friction

Now let's look at that ramp again ....

A box of mass M=10 kg sits on a ramp of angle theta=20 degrees. The coefficient of static friction between box and ramp is 0.30, and the coefficient of kinetic friction is 0.20.
1. Draw a set of coordinate axes which are tilted to match the ramp, so that the x-axis goes parallel to the slope and the y-axis goes perpendicular to it.
2. What is the normal force between the box and the ramp?
3. What is the magnitude of the force pulling the box down along the surface of the ramp?
4. What is the maximum possible force of friction on the box?
5. If you hold the box at the top of the ramp, then release it, does it slide downwards?
6. If not, how much must you increase the tilt of the ramp in order to make the box start to slide?
7. The ramp is tilted at theta=60 degrees. How fast does the box accelerate down the ramp?

Finding the uncertainty in the slope of a graph

First, read this description of finding the uncertainty in the slope of a line drawn through measurements:

How to find the uncertainty in a slope

Now, you try: here are some measurements of the cost to raise children. Note that the first child is the most expensive; subsequent kids can wear hand-me-down clothing, play with old toys, and so forth.

   Family       Number of children     Total cost (dollars)
  ---------------------------------------------------------
    Smith              3                   175,000
    Jones              2                   155,000
    Burbidge           7                   270,000
    Banner             3                   177,000
    Parker             5                   249,000
    Logan              4                   215,000
    Richards           4                   208,000
    Xavier             9                   352,000

Can you figure out the following?

the average cost of adding one more child to a family
the uncertainty in your estimate of this additional cost
the average fixed cost of raising a single child
the uncertainty in your estimate of this fixed cost