Found problems: 5
2010 F = Ma, 24
A uniform circular disk of radius $R$ begins with a mass $M$; about an axis through the center of the disk and perpendicular to the plane of the disk the moment of inertia is $I_\text{0}=\frac{1}{2}MR^2$. A hole is cut in the disk as shown in the diagram. In terms of the radius $R$ and the mass $M$ of the original disk, what is the moment of inertia of the resulting object about the axis shown?
[asy]
size(14cm);
pair O=origin;
pair
A=O,
B=(3,0),
C=(6,0);
real
r_1=1,
r_2=.5;
pen my_fill_pen_1=gray(.8);
pen my_fill_pen_2=white;
pen my_fill_pen_3=gray(.7);
pen my_circleline_draw_pen=black+1.5bp;
//fill();
filldraw(circle(A,r_1),my_fill_pen_1,my_circleline_draw_pen);
filldraw(circle(B,r_1),my_fill_pen_1,my_circleline_draw_pen);
// Ellipse
filldraw(yscale(.2)*circle(C,r_1),my_fill_pen_1,my_circleline_draw_pen);
draw((C.x,C.y-.75)--(C.x,C.y-.2), dashed);
draw(C--(C.x,C.y+1),dashed);
label("axis of rotation",(C.x,C.y-.75),3*S);
// small ellipse
pair center_small_ellipse;
center_small_ellipse=midpoint(C--(C.x+r_1,C.y));
//dot(center_small_ellipse);
filldraw(yscale(.15)*circle(center_small_ellipse,r_1/2),white);
pair center_elliptic_arc_arrow;
real gr=(sqrt(5)-1)/2;
center_elliptic_arc_arrow=(C.x,C.y+gr);
//dot(center_elliptic_arc_arrow);
draw(//shift((0*center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y-.2))*
(
yscale(.2)*
(
arc((center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y+2.4), .4,120,360+60))
),Arrow);
//dot(center_elliptic_arc_arrow);
// lower_Half-Ellipse
real downshift=1;
pair C_prime=(C.x,C.y-downshift);
path lower_Half_Ellipse=yscale(.2)*arc(C_prime,r_1,180,360);
path upper_Half_Ellipse=yscale(.2)*arc(C,r_1,180,360);
draw(lower_Half_Ellipse,my_circleline_draw_pen);
//draw(upper_Half_Ellipse,red);
// Why here ".2*downshift" instead of downshift seems to be not absolutely clean.
filldraw(upper_Half_Ellipse--(C.x+r_1,C.y-.2*downshift)--reverse(lower_Half_Ellipse)--cycle,gray);
//filldraw(shift(C-.1)*(circle((B+.5),.5)),my_fill_pen_2);//
filldraw(circle((B+.5),.5),my_fill_pen_2);//shift(C-.1)*
/*
filldraw(//shift((C.x,C.y-.45))*
yscale(.2)*circle((C.x,C.y-1),r_1),my_fill_pen_3,my_circleline_draw_pen);
*/
draw("$R$",A--dir(240),Arrow);
draw("$R$",B--shift(B)*dir(240),Arrow);
draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow);
draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow);
[/asy]
(A) $\text{(15/32)}MR^2$
(B) $\text{(13/32)}MR^2$
(C) $\text{(3/8)}MR^2$
(D) $\text{(9/32)}MR^2$
(E) $\text{(15/16)}MR^2$
2016 AMC 10, 24
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$
2009 F = Ma, 24
A uniform rectangular wood block of mass $M$, with length $b$ and height $a$, rests on an incline as shown. The incline and the wood block have a coefficient of static friction, $\mu_s$. The incline is moved upwards from an angle of zero through an angle $\theta$. At some critical angle the block will either tip over or slip down the plane. Determine the relationship between $a$, $b$, and $\mu_s$ such that the block will tip over (and not slip) at the critical angle. The box is rectangular, and $a \neq b$.
[asy]
draw((-10,0)--(0,0)--20/sqrt(3)*dir(150));
label("$\theta$",(0,0),dir(165)*6);
real x = 3;
fill((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle,grey);
draw((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle);
label("$a$",(0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x));
label("$b$",(3,3)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x),dir(60));
[/asy]
(A) $\mu_s > a/b$
(B) $\mu_s > 1-a/b$
(C) $\mu_s >b/a$
(D) $\mu_s < a/b$
(E) $\mu_s < b/a-1$
2008 F = Ma, 24
A ball is launched upward from the ground at an initial vertical speed of $v_\text{0}$ and begins bouncing vertically. Every time it rebounds, it loses a proportion of the magnitude of its velocity due to the inelastic nature of the collision, such that if the speed just before hitting the ground on a bounce is $v$, then the speed just after the bounce is $rv$, where $r < 1$ is a constant. Calculate the total length of time that the ball remains bouncing, assuming that any time associated with the actual contact of the ball with the ground is negligible.
(a) $\frac{2v_\text{0}}{g}\frac{1}{1-r}$
(b) $\frac{v_\text{0}}{g}\frac{r}{1-r}$
(c) $\frac{2v_\text{0}}{g}\frac{1-r}{r}$
(d) $\frac{2v_\text{0}}{g}\frac{1}{1-r^2}$
(e) $\frac{2v_\text{0}}{g}\frac{1}{1+(1-r)^2}$
2011 F = Ma, 24
A turntable is supported on a Teflon ring of inner radius $R$ and outer radius $R+\sigma$ ($\sigma<<R$), as shown in the diagram. To rotate the turntable at a constant rate, power must be supplied to overcome friction. The manufacturer of the turntable wishes to reduce the power required without changing the rotation rate, the weight of the turntable, or the coefficient of friction of the Teflon surface. Engineers propose two solutions: increasing the width of the bearing (increasing $\sigma$), or increasing the radius (increasing $R$). What are the effects of these proposed changes?
[asy]
size(200);
draw(circle((0,0),5.5),linewidth(2));
draw(circle((0,0),7),linewidth(2));
path arrow1 = (0,0)--5*dir(50);
draw(arrow1,EndArrow);
label("R",arrow1,NW);
draw((3,0)--(5.5,0),EndArrow);
path arrow2 = ((10,0)--(7,0));
draw(arrow2,EndArrow);
label("$\delta$",arrow2,N);
[/asy]
(A) Increasing $\sigma$ has no significant effect on the required power; increasing $R$ increases the required power.
(B) Increasing $\sigma$ has no significant effect on the required power; increasing $R$ decreases the required power.
(C) Increasing $\sigma$ increases the required power; increasing $R$ has no significant effect on the required power.
(D) Increasing $\sigma$ decreases the required power; increasing $R$ has no significant effect on the required power.
(E) Neither change has a significant effect on the required power.