Lab 2.1, Where a System Balances
The simulations are on this page. Work down the page and use them as you go. Parts 2, 3 and 4 ask you to predict something first, so write the prediction down before you touch the controls. Work symbolically first, write the limits on every integral, and take g = 9.80 m/s2. Show your work.
1. Warm-up: move the boundary and watch5 points
Two blocks joined by a rope, with a pull on the right one. The three buttons choose what you are calling the system, and the dashed line is the boundary. Nothing here is calculated.
Loading the simulation. If it does not appear, open it on the topic page.
Move the slider. Press each of the three buttons. Watch which arrows are drawn and which are not. Then write one sentence about what you noticed.5 pts
2. A rod that is heavier at one end15 points
A rod of length L along the x axis, from 0 to L, with density λ(x) = λ0(1 + kx/L). The panel gives both integrals in units that cancel λ0 and L, so you can compare them with your algebra.
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(a)Set k = 0. Find M and ∫ x dm, with limits, and then xcm. Check it against the panel. Now do the same for any k, and write xcm as one fraction in k and L. Check it at k = 3 and at k = 6.7 pts
(b)What does xcm approach as k grows without limit? Say in a sentence or two why it stops short of L. Then show that xcm is always less than 2L/3.8 pts
3. Removal is negative mass15 points
A 60 cm square steel plate. Press Corner bite, then Reset. The bite is then 24 cm by 24 cm, out of one corner.
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(a)Do the general case first. For a plate of side L with a square bite of side s, show that the center of mass moves c = s2 / [2(L + s)] along each axis. Then put in L = 60 cm and s = 24 cm and compare with the panel.7 pts
(b)The notch begins L/2 − s from the center. Use that to show the center of mass reaches the empty part when s2 + Ls − L2 = 0. Solve for s/L and name the number. Then find the crossing in the simulation and compare.8 pts
4. What actually follows the parabola15 points
A rod with a ball at each end, thrown across the room. Set the speed to 13.0 m/s, the angle to 52 degrees and the spin to 1.2 turns per second. Leave those three alone and change only the two masses.
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(a)First predict, in writing: if you change the two masses, does the time in the air change? Does the range? Then fill in the table. The length of the rod is never given anywhere; work it out from two rows. Finish with one sentence on what changed and what did not.15 pts
| Left ball (kg) | Right ball (kg) | Balance point from the left ball (m) | Time in the air (s) | Range (m) |
|---|---|---|---|---|
| 3 | 3 | |||
| 1 | 3 | |||
| 1 | 9 | |||
| 9 | 1 |