**UIUC Physics 211 James Scholars Problem Set 2**

In this assignment we will do some problems in 2-d motion and many-body systems in a bit more depth.

Problem 1

**Consider a bead of mass m that is free to
move around a horizontal, circular
ring of
wire (the wire passes through a hole in the bead). You may neglect
gravity in this problem
(assume the experiment is being done in space, far away from anything
else).
The radius of the ring of wire is r. The bead is given an
initial speed v_0 and it slides with a coefficient of friction mu_k.
In the subsequent
steps we will investigate the motion at later times. You should begin
by drawing
a free-body diagram at some instant of time. Note that there will be a
radial
acceleration, a_R, and a tangential acceleration, a_T,
in this problem.
**

1.1 Write Newton's 2nd law for the radial and tangential directions.

1.2 Combine the above equations to write a differential equation for *dv/dt*,
where
*v* is the speed at time *t*.

1.3 Solve the above differential equation to determine *v(t)*.
The solution
has the form *v = c1/(1+c2*t)* - find *c1* and *c2*.
Hint: if
*v_0 = 3* m/s, *mu_k = 0.1*, *r = 10* cm, and *t = 3*
s, *v(3) = 0.3* m/s.

Problem 2

**We will continue our analysis of Problem 1 with the bead.
**

2.1 Given your solution for *v(t)*, calculate the radial and
tangential components
of the acceleration, *a_R(t)* and *a_T(t)*, respectively.
From these calculate the total
acceleration *a_tot(t)*.

2.2 Given your solution for *v(t)*, write the differential
equation involving
*ds/dt*, where *s* is the position of the bead around the
circumference of the ring.
Solve this equation for *s(t)*.

2.3 Graph *s(t)*, *v(t)*, *a_R(t)*, and *a_T(t)*
for *v_0 = 3* m/s, *mu_k = 0.1*, *r = 10* cm
and let *t* vary from 0 to 10 s.

2.4 What is the speed, *v_1*, of the bead after 1 revolution of
the ring (using the
parameter values given in part 2.3)? What
is the speed *v_2* after 2 revolutions of the ring? (On your own
you may wish to calculate
these two speeds for the same bead with the same parameters, except
moving along a
straight wire.)

Problem 3

**Consider the Atwood's machine of Lecture 8. We wish to use this
machine to measure
our local acceleration of gravity with an accuracy of 5% [i.e.
(Delta g)/g = 0.05]. To begin, suppose we let the mass m_1
fall through a distance L.
**

3.1 Find an expression for the acceleration of gravity, *g*, in
terms of *m_1*, *m_2*, *L*
and *t*.

3.2 Now suppose we are able to measure time with an accuracy of *(Delta
t) = 0.1* s.
Assuming that, for example, *(Delta t)/t* can be approximated by
the differential *dt/t*,
write the relationship between *(Delta g)/g* and *(Delta t)/t*.
You can do this by starting
with the derivative *dg/dt* determined from the equation in the
previous part.

3.3 If *L = 3* m and *m_1 = 1* kg, determine the value of *m_2*
required to determine *g*
to 5%. If we want to measure *g* to 1% would the mass *m_2*
increase or decrease - why?
(On your own, you might want to consider the effect of the uncertainty
in the masses
of *m_1* and *m_2* on the determination of *g*.)