Found problems: 4
2008 F = Ma, 17
A mass $m$ is resting at equilibrium suspended from a vertical spring of natural length $L$ and spring constant $k$ inside a box as shown:
[asy]
//The Spring
import graph;
size(10cm);
guide coil(path g, real width=0.1, real margin = 1*width) {
real L = arclength(g);
real r = width / 2;
pair startpoint = arcpoint(g, margin);
real[][] isectiontimes = intersections(g, circle(c=startpoint,r=r));
real initialcirclecentertime = (isectiontimes.length == 1 ?
isectiontimes[0][0] : isectiontimes[1][0]);
pair startdir = dir(startpoint - point(g,initialcirclecentertime));
real startangle = atan2(startdir.y, startdir.x);
real startarctime = arclength(subpath(g, 0, initialcirclecentertime));
write(startarctime);
pair endpoint = arcpoint(g, L - margin);
real finalcirclecentertime = intersections(g, circle(c=endpoint,r=r))[0][0];
pair enddir = dir(endpoint - point(g,finalcirclecentertime));
real endangle = atan2(enddir.y, enddir.x);
real endarctime = arclength(subpath(g, 0, finalcirclecentertime));
write(endarctime);
real coillength = 2r;
real lengthalongcoils = L - 2*margin;
int numcoils = ceil(lengthalongcoils / coillength);
real anglesubtended = 2pi * numcoils - startangle + endangle;
real angleat(real arctime) {
return (arctime - startarctime) * (anglesubtended / (endarctime - startarctime)) + startangle;
}
pair f(real t) {
return arcpoint(g,t) + r * expi(angleat(t));
}
return subpath(g, 0, arctime(g, margin)) & graph(f, startarctime, endarctime, n=max(length(g), 20*numcoils+2), operator..) & subpath(g, arctime(g, L-margin), length(g));
}
draw(coil((0,0.25)--(0,1)));
//Outer Box
draw((-1,1)--(1,1),linewidth(2));
draw((-1,1)--(-1,-1.2),linewidth(2));
draw((-1,-1.2)--(1,-1.2),linewidth(2));
draw((1,1)--(1,-1.2),linewidth(2));
//Inner Box
draw((-0.2,0.25)--(0.2,0.25),linewidth(2));
path arc1=arc((-0.2,0.15),(-0.2,0.25),(-0.3,0.15));
path arc2=arc((0.2,0.15),(0.3,0.15),(0.2,0.25));
draw(arc1,linewidth(2));
draw(arc2,linewidth(2));
draw((-0.3,0.15)--(-0.3,-0.3),linewidth(2));
draw((0.3,0.15)--(0.3,-0.3),linewidth(2));
path arc3=arc((-0.2,-0.3),(-0.3,-0.3),(-0.2,-0.4));
draw(arc3,linewidth(2));
path arc4=arc((0.2,-0.3),(0.2,-0.4),(0.3,-0.3));
draw((-0.2,-0.4)--(0.2,-0.4),linewidth(2));
draw(arc4,linewidth(2));
[/asy]
The box begins accelerating upward with acceleration $a$. How much closer does the equilibrium position of the mass move to the bottom of the box?
(a) $(a/g)L$
(b) $(g/a)L$
(c) $m(g + a)/k$
(d) $m(g - a)/k$
(e) $ma/k$
2010 F = Ma, 17
Four masses $m$ are arranged at the vertices of a tetrahedron of side length $a$. What is the gravitational potential energy of this arrangement?
(A) $-2\frac{Gm^2}{a}$
(B) $-3\frac{Gm^2}{a}$
(C) $-4\frac{Gm^2}{a}$
(D) $-6\frac{Gm^2}{a}$
(E) $-12\frac{Gm^2}{a}$
2011 F = Ma, 17
Jonathan applies a normal force that is just enough to keep the rope from slipping. Becky makes a small jump, barely leaving contact with the floor of the box. Upon landing on the box, the force of the impact causes the rope to start slipping from Jonathan’s hand. At what speed does the box smash into the ground? Assume Jonathan’s normal force does not change.
(A) $\sqrt{2gH}(\mu_k/\mu_s)$
(B) $\sqrt{2gH}(1-\mu_k/\mu_s)$
(C) $\sqrt{2gH}\sqrt{\mu_k/\mu_s}$
(D) $\sqrt{2gH}\sqrt{1-(\mu_k/\mu_s)}$
(E) $\sqrt{2gH}(\mu_s-\mu_k)$
2009 F = Ma, 17
You are given a standard kilogram mass and a tuning fork that is calibrated in Hz. You are also provided with a complete collection of laboratory equipment, but none of it is calibrated in SI units. You do not know the values of any fundamental constants. Which of the following quantities could you measure in SI units?
(A) The acceleration due to gravity.
(B) The speed of light in a vacuum.
(C) The density of room temperature water.
(D) The spring constant of a given spring.
(E) The air pressure in the room.