TA: Tomoyuki Nakayama
Tuesday, April 5, 2011
PHY 2048: Physic 1, Discussion Section 3706H
Quiz 9 (Homework Set #11)
Name:
UFID:
Formula sheets are not allowed. Do not store equations in your calculator. You have to solve
problems on your own; memorizing final algebraic expressions from homework assignments and
just plugging numbers into them will not give you full credit. Leave all your work.
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Three identical stars of mass M = 8.00 × 1030 kg form an equilateral triangle that rotates around the
triangle's center as the stars move in a common circle about that center. The triangle has edge
length L= 3.00 × 1010 m.
a) What is the speed of the stars?
By symmetry, all stars have the same speed. Thus it is enough to consider the motion of one of the
stars. We label them star 1, 2 and 3. Star 1 rotates in a circle. Therefore, it accelerates toward the
center with magnitude v2/R, where R is the radius of the circular orbit. The radius of the circle and
the edge length of the triangle are related as
Rcosθ = L/2 ⇒ R = L/(2cosθ),
where θ = 30º. The centripetal acceleration is provided by two gravitational forces from other 2
stars. The two forces have the same magnitude and each makes an angle θ = 30º to the radial
direction. Newton’s 2nd law yields
M(v2/R) = F1 ⇒ M(v2/(L/(2cosθ))) = F12cosθ + F13cosθ = 2(Gm2/L2)cosθ
⇒ v = √(Gm/L) = 1.334 × 105 m/s
b) What is the period of the circular motion of the stars?
The period of the circular motion is
T = 2πR/v = 2π(L/(2cosθ))/v = 8.16 × 105 s