Here are three expressions that you may or may not need.
The planar moment of inertia of a rigid body about a point P on the body is $I_P = I_{CM} + md^2$,
where $I_{CM}$ is the moment of inertia about the body's center of mass and $d$ is the distance between
P and the center of mass.
The angular momentum balance about a point P on a rigid body can be written as
$M_P = \dot{H}_{CM} + r_{C/P} \times ma_{CM}$ or as $M_P = I_P \ddot{\theta} \hat{k} + r_{C/P} \times ma_P$.
1) Hinged rigid body falling onto surface (100 pts)
Consider a rigid body with a fixed point O on a hinge that it is free to rotate around (there is
no way to have a local reaction moment at O, but there can be a reaction force R). The body's
center of mass is at point C, at a distance $l$ from O. The body has total mass $m$. The moment of
inertia of the body about the center of mass is $I_C$.
We refer to two right handed frames of orthogonal unit vectors: $\hat{b}_1$ and $\hat{b}_2$ are attached to
(rotate with) the body, with $\hat{b}_1$ in the direction between O and C. The vectors $\hat{t}$ and $\hat{n}$ are fixed,
tangential and normal to a horizontal surface passing through O. A unit vector $\hat{k}$ (not shown)
points out of the plane.
Gravity points down. The angle $\theta$ is measured counterclockwise from the horizontal surface.
In your answers, write angular velocities and accelerations using $\theta$ and its time derivatives $\dot{\theta}$ and $\ddot{\theta}$.
Evaluate any dot and cross products unless told not to.
"Write u in the $\hat{a}_1$-$\hat{a}_2$ frame" means write an expression of the form (something)$\hat{a}_1$+(somethingelse)$\hat{a}_2$.
a) Draw a free-body diagram showing all of the forces on the body. 5 pts
b) Write the angular velocity of the body. 5 pts
c) Write its center of mass velocity $v_C$ in the $\hat{b}_1$-$\hat{b}_2$ frame. 5 pts
d) Take a time derivative and write its center of mass acceleration $a_C$ in the same frame. 5 pts
e) Write the linear momentum balance for the body (you may leave in general form). 10 pts
f) Find the components of the reaction force at O in the $\hat{t}$ and $\hat{n}$ directions, in terms of functions
of $\theta$ and its derivatives. 15 pts
g) Write the angular momentum balance for the body around its center of mass (you may leave
in general form, without evaluating cross products). 10 pts
h) Consider all vectors in the plane, including the reaction force at O, as having components
in the radial and azimuthal directions $\hat{b}_1$ and $\hat{b}_2$. Compute the sum of moments of forces about
the body's center of mass, expressed using these components. (Your result should be in the $\hat{k}$
direction.) 10 pts
i) Find the azimuthal reaction force component, in terms of functions of $\theta$ and its derivatives.
(This should be a simple relationship). 10 pts
j) Write the kinetic energy of the body in terms of its center of mass speed $v_C$ and $\dot{\theta}$. 10 pts
k) Rewrite in terms of $\dot{\theta}$ alone. 5 pts
l) Rewrite further in terms of $I_O$, the moment of inertia of the body about point O. 5 pts
m) The body is dropped from rest, with a potential energy of $PE_i$ (we could write this in terms
of $\theta$, but we won't do this). When it hits the surface ($\theta = 0$, say), it has a potential energy of zero.
How fast is it rotating (what is $|\dot{\theta}|$)? 5 pts
