Found problems: 4
2009 F = Ma, 22
Determine the period of orbit for the star of mass $3M$.
(A) $\pi \sqrt{\frac{d^3}{GM}}$
(B) $\frac{3\pi}{4}\sqrt{\frac{d^3}{GM}}$
(C) $\pi \sqrt{\frac{d^3}{3GM}}$
(D) $2\pi \sqrt{\frac{d^3}{GM}}$
(E) $\frac{\pi}{4} \sqrt{\frac{d^3}{GM}}$
2011 F = Ma, 22
This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range.
[asy]
size(200);
draw((0,0)--(10,0)--(10,7)--(0,7)--cycle);
for (int i=1;i<10;++i) {
draw((i,0)--(i,7),dashed+linewidth(0.5));
}
for (int j=1;j<7;++j) {
draw((0,j)--(10,j),dashed+linewidth(0.5));
}
draw((0,0)--(0,-0.3));
draw((4,0)--(4,-0.3));
draw((8,0)--(8,-0.3));
draw((0,0)--(-0.3,0));
draw((0,2)--(-0.3,2));
draw((0,4)--(-0.3,4));
draw((0,6)--(-0.3,6));
label("0",(0,-0.5),S);
label("1000",(4,-0.5),S);
label("2000",(8,-0.5),S);
label("0",(-0.5,0),W);
label("10",(-0.5,2),W);
label("20",(-0.5,4),W);
label("30",(-0.5,6),W);
label("I",(1,-1.5),S);
label("II",(6,-1.5),S);
label("III",(9,-1.5),S);
label(scale(0.95)*"Engine Revolutions per Minute",(5,-3.5),N);
label(scale(0.95)*rotate(90)*"Output Torque (Nm)",(-1.5,3),W);
path A=(0.9,2.7)--(1.213, 2.713)--
(1.650, 2.853)--
(2.087, 3)--
(2.525, 3.183)--
(2.963, 3.471)--
(3.403, 3.888)--
(3.823, 4.346)--
(4.204, 4.808)--
(4.565, 5.277)--
(4.945, 5.719)--
(5.365, 6.101)--
(5.802, 6.298)--
(6.237, 6.275)--
(6.670, 6.007)--
(7.101, 5.600)--
(7.473, 5.229)--
(7.766, 4.808)--
(8.019, 4.374)--
(8.271, 3.894)--
(8.476, 3.445)--
(8.568, 2.874)--
(8.668, 2.325)--
(8.765, 1.897)--
(8.794, 1.479)--(8.9,1.2);
draw(shift(0.1*right)*shift(0.2*down)*A,linewidth(3));
[/asy]
At what engine RPM (revolutions per minute) does the engine produce maximum power?
(A) $\text{I}$
(B) At some point between $\text{I}$ and $\text{II}$
(C) $\text{II}$
(D) At some point between $\text{II}$ and $\text{III}$
(E) $\text{III}$
2010 F = Ma, 22
A balloon filled with helium gas is tied by a light string to the floor of a car; the car is sealed so that the motion of the car does not cause air from outside to affect the balloon. If the car is traveling with constant speed along a circular path, in what direction will the balloon on the string lean towards?
[asy]
size(300);
draw(circle((0,0),7));
path A=(1,2)--(1,-2)--(-1,-2)--(-1,2)--cycle;
filldraw(shift(7*left)*A,lightgray);
draw((-7,0)--(-7,5),EndArrow(size=21));
label(scale(1.5)*"A",(-8,2),2*N);
label(scale(1.5)*"B",(-8,0),2*W);
label(scale(1.5)*"C",(-7,-2),3*S);
label(scale(1.5)*"D",(-6,0),2*E);
[/asy]
(A) A
(B) B
(C) C
(D) D
(E) Remains vertical
2008 F = Ma, 22
A bullet of mass $m_\text{1}$ strikes a pendulum of mass $m_\text{2}$ suspended from a pivot by a string of length $L$ with a horizontal velocity $v_\text{0}$. The collision is perfectly inelastic and the bullet sticks to the bob. Find the minimum velocity $v_\text{0}$ such that the bob (with the bullet inside) completes a circular vertical loop.
(a) $2\sqrt{Lg}$
(b) $\sqrt{5Lg}$
(c) $(m_\text{1}+m_\text{2})2\sqrt{Lg}/m_\text{1}$
(d) $(m_\text{1}-m_\text{2})\sqrt{Lg}/m_\text{2}$
(e) $(m_\text{1}+m_\text{2})\sqrt{5Lg}/m_\text{1}$