Student Name_______________________,
Panther ID______________
PHY4222 HMWK set 3
1. The following object, consisting an arc (corresponding to radius of R), and equal
length (R) two straight wires, has a uniform linear density, and is in the x-y plane. The
arc spans an angel of θ. The total mass is M.
a) What is the moment of inertial about an axis that is along z-­‐direction and goes through the origin O? b) Using the results from a similar (but not identical !) earlier HMWK problem, what is the distance between the center-­‐of-­‐mass and the origin? c) If this object undergoes small oscillation about the axis described in a), what is the period?
2. An equilateral triangular plate has a mass of M, the side length is R. What is the
center-of-mass moment of Inertia, Icm, Expressed in terms of M and R? (The rotational
axis is z axis, coming out of the paper perpendicularly, from the center)
Hint: Make use of symmetry and parallel axis theorum, and you can turn this problem
into some simple algebra. Obviously, you could have done the calculation using integrals.
But it is not allowed for this problem.
Icm = cMR 2
c =?
M
R
€
3. A hollow cylinder has an outer radius of R, and inner radius of R0, is at rest on a flat
and frictionless surface. It was found that, when a horizontal force strikes at 3R/4 above
the center, the cylinder will roll without slipping. Determine the value of R0/R.
4. A cube, with side length of b, is balanced (in unstable equilibrium) on one of its edge
on a horizontal plane. A small disturbance tips it over. Determine the cube’s angular
velocity ω, as a function of θ (as shown in the graph); You should consult the lecture
notes if you need some hints.
a) ω(θ)=? if there is friction so that the block doesn’t slip?
b) ω(θ)=? if there is no friction so that the block does slip?
(For part c) and d), assume a very small initial angle of θ0, right after it started to fall due
to the disturbance)
c) how long does it take for the cube to fall flat, if there IS friction, and no slipping?
d) how long does it take for the cube to fall flat, if there is NO friction?