## Problems

### practice

- Write something.
- Write something else..
- The photographs below show a counterweighted, steel drawbridge when the span is down and when the span is up.
Determine the following quantities in terms of the weight of the bridge span.

- The weight of the counterweight
*and*the tension when the span is down. - The tension when the bridge is open
*and*the torque needed to keep the span up.

- The weight of the counterweight
- Write something completely different.

### conceptual

- A hammer is balanced on one finger as shown in the diagram to the right. This divides the hammer up into two parts: the handle on the left and the head (plus a little bit of the handle) on the right. Which part weighs more — or are they both equally heavy? Justify your answer.
- How can all three types of equilibrium — stable, unstable, and neutral — be demonstrated using…
- an egg
- a cone
- a torus (a shape like a donut or a bagel)

- Explain the reasoning behind each of the following general rules of design.
- Aircraft carriers are designed to be stable in the ocean.
- Fighter planes are designed to be unstable in flight.

### worksheets

- The Physics Teacher has published several articles containing free body diagram worksheets. They are available free to members of the American Association of Physics Teachers (AAPT). Everyone else has to pay.
- Free-body diagrams revisited—II. James E. Court.
*The Physics Teacher*. Vol 37 No. 8 (November 1999): 490–495.- RE1–RE16: Rotational Equilibrium
- RN1–RN9: Rotational Nonequilibrium

- Free-body diagrams revisited—II. James E. Court.

### statistical

- center-of-population.txt

The center of population of the United States as defined by the Census Bureau is the same as that of the center of gravity of a collection of point masses on a plane. It is the point at which a weightless, smooth, spherical shell in the shape of the "lower 48" states and the District of Columbia would balance if weights of identical size were placed on it — each weight representing the location of one person. On such an imaginary surface, north-south distances between parallels of*latitude*(ϕ) are identical and their angular measure in degrees may be used as units of displacement. In contrast, east-west distances between meridians of*longitude*(λ) are not constant but vary with latitude from a maximum at the equator to zero at the poles. Multiplying by the cosine of the latitude will correct for this convergence of the meridians at the poles. In addition, small areas of the country are used as data points rather than individual human beings, which reduces the computational burden. (In 1960 43,000 areas were used but by 2000, this number had risen to more than 8,000,000. By 2020 or 2030, the number of areas will probably equal the number of residents.)

Thus, the center of population of the US computed by the Census Bureau is the point whose latitude (φ̅) and longitude (λ̅) satisfy the equations…φ̅ = ∑ *w*φ_{i}_{i}λ̅ = ∑ *w*_{i}λcos φ_{i}_{i}∑ *w*_{i}∑ *w*cos φ_{i}_{i}, λ_{i}, and_{i}*w*are the latitude, longitude, and population of the census areas included in the calculation._{i}- The data on the accompanying tabs-delimited text file give the population and the effective latitude and longitude in degrees of the fifty states and the District of Columbia from the 2000 census. Using this data, determine the coordinates of the population center of the United States at this time. (Be sure to exclude Alaska and Hawaii from your calculations, but do include the District of Columbia.)
- In what state is this point located? Which county? What is the nearest incorporated community (city, village, or town)? What is the nearest street intersection?
- Go to this location and await further instructions.

### investigative

- Determine the mass of a ruler using a known weight to balance it over a pivot.