AP Physics – Momentum and Systems of Particles Example Problems
1. Determine
the position of the center of mass for the following set of masses:
m_{1} = 200.0 g, r_{1} = (25.0 i + 13.0 j) cm
m_{2} = 300.0 g, r_{2} = (15.0 i
+ 8.00 j) cm
m_{3} = 900.0 g, r_{3} = (5.00 i  10.0 j) cm
2. Suppose a fourth mass is added to the set of masses in the previous example so that the center of mass is relocated to the origin. (a) Show that this fourth mass must lie on a line with equation y = 0.8x. (b) What would be the value of the fourth mass if it is located at r_{4} = (5.00 i + 4.00 j) cm?
3. Determine the center of mass of a semicircle of radius R made of a thin sheet of material.
4. Determine the center of mass of a thin sheet of material bound by: y £ x^{2}, y ³ 0, x £ 2.
5. A man, mass 90.0 kg, and a woman, who is lighter, are seated at rest in a 20.0 kg canoe that floats upon a placid frictionless lake. The seats are 2.80 m apart and are symmetrically located on each side of the canoe’s center of mass. The man and woman decide to swap seats and the man notices that the canoe moves 30.0 cm relative to a submerged log during the exchange. The man uses this fact to determine the woman’s mass (the guy is a physics nerd – he has his trusty calculator in his pocket and hey, he can’t think of anything to say to the woman anyway). (a) What is the woman’s mass? (b) Will the nerd completely ruin the date by showing the woman his calculations?
6. Consider a planet orbiting the sun. There exist five “Lagrange points” at which a third body, such as an asteroid, can orbit in sync with the planet. In other words the third body will have the same period as the planet and maintain its position relative to the planet as it orbits the sun. This is possible due to the mutual gravity of the sun and the planet on the third body. Two of these Lagrangian points, L_{4} and L_{5}, form perfect equilateral triangles with the Sun and the planet. Show that an object at L_{4} or L_{5} will indeed orbit in sync with the planet.
7. Graph the Sun’s “wobble” due to the planets’ gravity. Make the simplifying assumption that each planet moves in a circular orbit at a constant speed (all in the same plane) about the solar system’s center of mass. The Sun moves about this center of mass as well. Use parametric equations of the form x(t) = r cos(w t + d) and y(t) = r sin(w t + d) to model the motion of each planet. Let t be measured in years elapsed since Jan. 1, 1960. The chart below shows information needed, including phase angles that specify the positions at t = 0. Can you show that Mercury, Mars, and Pluto have a negligible effect on this problem? Note: this exercise is based upon and inspired by a diagram in the May 2003 Sky & Telescope.

Mass (kg) 
Orbit radius (AU) 
Period (yrs) 
Phase Angle 
Sun 
1.99 × 10^{30} 
n/a 
n/a 
n/a 
Venus 
4.87 × 10^{24} 
0.723 
0.6152 
175° 
Earth 
5.974 × 10^{24} 
1.000 
1.000 
100° 
Jupiter 
1.90 × 10^{27} 
5.20 
11.86 
255° 
Saturn 
5.68 × 10^{26} 
9.54 
29.42 
280° 
Uranus 
8.69 × 10^{25} 
19.19 
83.75 
139° 

1.02 × 10^{26} 
30.07 
163.7 
217° 
Answers:
1. r_{cm} = (3.57 i  2.86 j) cm
2. a.
b. m_{4} = 1000 g
3. 4R/3π from the center of the circle
4. x_{cm} = 1.5, y_{cm} = 1.2
5. a. 70.6 kg
b. um, yes.
6. Hint: All three bodies move in circles of different radii about the system center of mass while moving “in formation” of an equilateral triangle. The Sun moves in the smallest circle and the asteroid will move in the largest circle.
7. Here
is a plot of the Sun’s wobble during the years 1960 through 2025. The scales give values in AU. The green curve is the path of the Sun’s
center and the yellow circle represents the Sun at
correct scale in its final location during this interval.