Physics 171.101   Exam 2   Nov, 17, 1999
Prof. Barnett

1.

The adjacent figure shows two students sitting on a massless,
5 meter long plank. Rashidat Agbaje, M_R = 60 kg, is on the
extreme right end of the plank, and Lucas Finn, M_L = 60 kg,
is on the extreme left end of the plank. The plank is suspended
from a wire 2 meters from the left end. Take |g| = 10 m/sec² and
consider both students to be ``point masses'' throughout
this problem.

A)

(9 pts) What is the torque about the point P where the wire connects to the plank?

B)

(9 pts) What is the angular acceleration, $\vec{\alpha}$, of the plank about point P?

C)

(9 pts) What is the acceleration, $\vec{a}_R$, of Rashidat?

D)

(9 pts) What is the acceleration, $\vec{a}_{CM}$, of the center of mass of Rashidat and Lucas?

E)

(9 pts) What is the tension of the wire?

2.

As shown in the adjacent figure, a six meter long plank
is attached vertically to a frictionless axle through its
center. The total mass of plank is 30 kg and its moment of
inertia about the axle is 90 kg m². A point-mass student,
Anna Widmer, M_A, mass = 50 kg, runs past at a speed of
5 m/sec and grabs onto the bottom end of the plank. After
she has grabbed the plank she and the plank rotate about the
axle with an initial instantaneous angular velocity $\vec{\omega}_0$. Anna
rises into the air, holding onto the plank, and eventually stops at some height, h, above the ground when the plank reaches some maximum angle, $\theta_{max}$.

A)

(7 pts) Is there some point, P, in the ``plank plus Anna'' system about which there is no torque at the instant when Anna grabs the plank? If ``Yes''. explain where the point, P, is and why there is no torque.

B)

(8 pts) What are 1) Anna's momentum and 2) angular momentum about the point P at the instant before she grabs the plank?

C)

(8 pts) After Anna grabs the plank, what is the total moment of inertia of Anna and the plank about the point P?

D)

(8 pts) What is the angular velocity, $\vec{\omega}_0$, of Anna and the plank about the axle immediately after she grabs the plank?

E)

(8 pts) What is the total mechanical energy of Anna plus the plank at the instant after she grabs the plank?

F)

(8 pts) Find the maximum height, h, that Anna reaches above the ground due to the rotation of the plank about the axle to its maximum angle, $\theta_{max}$.

G)

(8 pts) Did the linear momentum of the ``Anna plus plank'' system change when Anna grabbed the plank? If ``Yes'', what impulse was exerted on the system??

You can take $\pi = 3.0$ if you want.

You can take |g| = 10 meters/seconds² if you want.

$\vec{v}(t) = \vec{v}(0) + \vec{a}t$

$x(t) = x(0) + v(0)t + \frac{1}{2} at^2$

v²(t) = v²(0) + 2 a (x(t) - x(0))

W = mg

a = v²/r

$\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z = a~ b~ cos(\theta)$

$\vert\vec{a} \times \vec{b}\vert = a~ b~ sin(\theta)$

F=$\mu$ N

$\vec{F} = m \vec{a}$

$\vec{v} = \vec{\omega} \times \vec{r}$

$ax^2 + bx + c = 0 \rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

T = 1/f

$\omega = \sqrt{k/m}$

KE = $\frac{1}{2} m v^2$

KE = $\frac{1}{2} I \omega^2$

$\vec{p} = m\vec{v}$

PE = mgh

$\vec{L} = I\vec{\omega}$

$\vec{a} = \vec{\alpha} \times \vec{R}$

I = MR²

I = I_CM + MR²

$\vec{\tau} = \vec{R} \times \vec{F}$

$\vec{\tau} = I \vec{\alpha}$

$\vec{L} = \vec{R} \times \vec{P}$

$M_{total} \vec{R}_{CM} = \Sigma (M_i \vec{R}_i)$

$W = \vec{F} \cdot \vec{R}$

$P = \frac{dW}{dt} = \vec{F} \cdot \vec{V}$

$\vec{I} = \int \vec{F} \cdot dt = \Delta \vec{P}$