Found problems: 4
2011 F = Ma, 23
A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle), and that the gravitation due to a uniform sphere is identical to that of a point mass. Which of the following best explains the discrepancy?
(A) Because the experiment takes place very close to the surface of the sphere, it is no longer valid to replace the sphere with a point mass.
(B) Because the particle strikes the ground, it is not in orbit of the planet and therefore can follow a nonelliptical path.
(C) Kate disregarded the fact that motions around a point mass may also be parabolas or hyperbolas.
(D) Kepler’s laws only hold in the limit of large orbits.
(E) The path is an ellipse, but is very close to a parabola due to the short length of the flight relative to the distance from the center of the planet.
2008 F = Ma, 23
Consider two uniform spherical planets of equal density but unequal radius. Which of the following quantities is the same for both planets?
(a) The escape velocity from the planet’s surface.
(b) The acceleration due to gravity at the planet’s surface.
(c) The orbital period of a satellite in a circular orbit just above the planet’s surface.
(d) The orbital period of a satellite in a circular orbit at a given distance from the planet’s center.
(e) None of the above.
2010 F = Ma, 23
Two streams of water flow through the U-shaped tubes shown. The tube on the left has cross-sectional area $A$, and the speed of the water flowing through it is $v$; the tube on the right has cross-sectional area $A'=1/2A$. If the net force on the tube assembly is zero, what must be the speed $v'$ of the water flowing through the tube on the right?
Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit.
[asy]
// Code by riben
size(300);
draw(arc((0,0),10,90,270));
draw(arc((0,0),7,90,270));
draw((0,10)--(25,10));
draw((0,-10)--(25,-10));
draw((0,7)--(25,7));
draw((0,-7)--(25,-7));
draw(ellipse((25,8.5),0.5,1.5));
draw(ellipse((25,-8.5),0.5,1.5));
draw((20,8.5)--(7,8.5),EndArrow(size=7));
draw((7,-8.5)--(20,-8.5),EndArrow(size=7));
draw(arc((-22,0),12,90,-90));
draw(arc((-22,0),7,90,-90));
draw((-22,12)--(-42,12));
draw((-22,-12)--(-42,-12));
draw((-22,7)--(-42,7));
draw((-22,-7)--(-42,-7));
draw(ellipse((-42,9.5),1.5,2.5));
draw(ellipse((-42,-9.5),1.5,2.5));
draw((-38,9.5)--(-23,9.5),EndArrow(size=7));
draw((-23,-9.5)--(-38,-9.5),EndArrow(size=7));
[/asy]
(A) $1/2v$
(B) $v$
(C) $\sqrt{2}v$
(D) $2v$
(E) $4v$
2009 F = Ma, 23
A mass is attached to an ideal spring. At time $t = \text{0}$ the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is $T$. At what time is the power delivered [i]to[/i] the mass by the spring first a maximum?
(A) $t = \text{0}$
(B) $t = T/\text{8}$
(C) $t = T/\text{4}$
(D) $t = \text{3}T/\text{8}$
(E) $t = T/\text{2}$