Found problems: 4
2008 F = Ma, 18
A uniform circular ring of radius $R$ is fixed in place. A particle is placed on the axis of the ring at a distance much greater than $R$ and allowed to fall towards the ring under the influence of the ring’s gravity. The particle achieves a maximum speed $v$. The ring is replaced with one of the same (linear) mass density but radius $2R$, and the experiment is repeated. What is the new maximum speed of the particle?
(a) $\frac{1}{2}v$
(b) $\frac{1}{\sqrt{2}}v$
(c) $v$
(d) $\sqrt{2}v$
(e) $2v$
2009 F = Ma, 18
A simple pendulum of length $L$ is constructed from a point object of mass $m$ suspended by a massless string attached to a fixed pivot point. A small peg is placed a distance $2L/3$ directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced $5$ degrees from the vertical and released. How long does it take to return to its starting position?
[asy]
// Code by riben
size(275);
draw(circle((0,0),1),linewidth(2));
filldraw(circle((0,0),1),gray);
draw((0,0)--(0,-70.8));
draw(circle((0,-71.8),3));
filldraw(circle((0,-71.8),3),gray);
draw(circle((0,-45),1));
filldraw(circle((0,-45),1),gray);
filldraw(circle((15,-70),3),gray,linewidth(0.2));
filldraw(circle((-15,-67),3),gray,linewidth(0.2));
draw((0,0)--(14.5,-66.5),dashed);
draw((0,-45)--(-13,-65),dashed);
// Labels
label("Fixed Pivot Point",(0,0),4*E);
label("Small Peg",(0,-45),12*E);
label("Point Object of mass m",(0,-70),17*E);
draw((-40,1)--(-40,-76.8),EndArrow(size=5));
draw((-40,-76.8)--(-40,1),EndArrow(size=5));
label("L",(-40,-37.9),E*2);
[/asy]
(A) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{\frac{2}{3}}\right)$
(B) $\pi \sqrt{\frac{L}{g}} \left(2+\frac{2}{\sqrt{3}}\right)$
(C) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{3}\right)$
(D) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{3}\right)$
(E) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{\sqrt{3}}\right)$
2010 F = Ma, 18
Which of the following represents the force corresponding to the given potential?
[asy]
// Code by riben
size(400);
picture pic;
// Rectangle
draw(pic,(0,0)--(22,0)--(22,12)--(0,12)--cycle);
label(pic,"-15",(2,0),S);
label(pic,"-10",(5,0),S);
label(pic,"-5",(8,0),S);
label(pic,"0",(11,0),S);
label(pic,"5",(14,0),S);
label(pic,"10",(17,0),S);
label(pic,"15",(20,0),S);
label(pic,"-2",(0,2),W);
label(pic,"-1",(0,4),W);
label(pic,"0",(0,6),W);
label(pic,"1",(0,8),W);
label(pic,"2",(0,10),W);
label(pic,rotate(90)*"F (N)",(-2,6),W);
label(pic,"x (m)",(11,-2),S);
// Tick Marks
draw(pic,(2,0)--(2,0.3));
draw(pic,(5,0)--(5,0.3));
draw(pic,(8,0)--(8,0.3));
draw(pic,(11,0)--(11,0.3));
draw(pic,(14,0)--(14,0.3));
draw(pic,(17,0)--(17,0.3));
draw(pic,(20,0)--(20,0.3));
draw(pic,(0,2)--(0.3,2));
draw(pic,(0,4)--(0.3,4));
draw(pic,(0,6)--(0.3,6));
draw(pic,(0,8)--(0.3,8));
draw(pic,(0,10)--(0.3,10));
draw(pic,(2,12)--(2,11.7));
draw(pic,(5,12)--(5,11.7));
draw(pic,(8,12)--(8,11.7));
draw(pic,(11,12)--(11,11.7));
draw(pic,(14,12)--(14,11.7));
draw(pic,(17,12)--(17,11.7));
draw(pic,(20,12)--(20,11.7));
draw(pic,(22,2)--(21.7,2));
draw(pic,(22,4)--(21.7,4));
draw(pic,(22,6)--(21.7,6));
draw(pic,(22,8)--(21.7,8));
draw(pic,(22,10)--(21.7,10));
// Paths
path A=(0,6)--(5,6)--(5,4)--(11,4)--(11,8)--(17,8)--(17,6)--(22,6);
path B=(0,6)--(5,6)--(5,2)--(11,2)--(11,10)--(17,10)--(17,6)--(22,6);
path C=(0,6)--(5,6)--(5,5)--(11,5)--(11,7)--(17,7)--(17,6)--(22,6);
path D=(0,6)--(5,6)--(5,7)--(11,7)--(11,5)--(17,5)--(17,6)--(22,6);
path E=(0,6)--(5,6)--(5,8)--(11,8)--(11,4)--(17,4)--(17,6)--(22,6);
draw(A);
label("(A)",(9.5,-3),4*S);
draw(shift(35*right)*B);
label("(B)",(45.5,-3),4*S);
draw(shift(20*down)*C);
label("(C)",(9.5,-23),4*S);
draw(shift(35*right)*shift(20*down)*D);
label("(D)",(45.5,-23),4*S);
draw(shift(40*down)*E);
label("(E)",(9.5,-43),4*S);
add(pic);
picture pic2=shift(35*right)*pic;
picture pic3=shift(20*down)*pic;
picture pic4=shift(35*right)*shift(20*down)*pic;
picture pic5=shift(40*down)*pic;
add(pic2);
add(pic3);
add(pic4);
add(pic5);
[/asy]
2011 F = Ma, 18
A block of mass $\text{m = 3.0 kg}$ slides down one ramp, and then up a second ramp. The coefficient of kinetic friction between the block and each ramp is $\mu_\text{k} = \text{0.40}$. The block begins at a height $\text{h}_\text{1} = \text{1.0 m}$ above the horizontal. Both ramps are at a $\text{30}^{\circ}$ incline above the horizontal. To what height above the horizontal does the block rise on the second ramp?
(A) $\text{0.18 m}$
(B) $\text{0.52 m}$
(C) $\text{0.59 m}$
(D) $\text{0.69 m}$
(E) $\text{0.71 m}$