1. Given: a solid cylinder with radius R and height H. The volume mass density ρ is not constant,
but varies with distance r to the central axis as follows: ρ = ρ0 (R-r)2
a) Calculate the mass of the cylinder
b) Calculate the moment of inertia with respect to the central axis
The simplified answer should come out as follows:
I = (1/5)MR2
2. A hollow cylinder of height H has a total mass M which is uniformly distributed. The cylinder has
a hole in the center with radius R1. The outer radius of the cylinder is R2. Show that the moment
of inertia of this object with respect to the central axis (perpendicular to the disk) is given by the
formula:
1
𝐼 = 𝑀(𝑅1 2 + 𝑅2 2 )
2
Hint: split up the cylinder in concentric shells of infinitesimal thickness as shown in the example
in the lecture. Use the symbol ρ for the volume mass density (in kg/m3). Calculate the mass dm
in the infinitesimal shell. Use this expression to calculate both the mass and the moment of
inertia, and compare the two expressions.
3. On a harsh winter day, a 1500 kg vehicle takes a circular banked exit ramp (radius R = 150 m;
banking angle θ= 10 degrees) at a speed of 30 mph, since the speed limit is 35 mph. However,
the exit ramp is completely iced up (= frictionless). To make matters worse, a wind is blowing
parallel to the ramp in a downward direction. The wind exerts a force of 3000 N. Under these
conditions, can the driver continue to follow a safe horizontal circle on the exit ramp and stay
below the speed limit?
To convert mph into m/s use 1 mi = 1607 m and 1 hr is 3600 s
4. Using Newton’s universal law of gravity, calculate the orbital speed of a planet m around a star
M, and show that the time T to go around the star is related to the orbital radius R of the planet
according to: T2 = R3.
5. Two workers are holding a thin plate with length 5 m and height 2 m at rest by supporting the
plate in the bottom corners. The workers are standing at rest on a slope of 10 degrees. Treat
these supporting forces as vertical normal forces. Calculate their magnitudes to see if both
workers are sharing “the job” fairly.
6. An 80 kg block is placed 2 m away from the endpoint of a horizontal steel beam of length 6.6 m
and mass 1,450 kg. The plank makes contact with a vertical wall on one end, and the other
endpoint is attached to a massless cable that makes an angle of 30 degrees with the horizontal
and ties into the vertical wall as well. Calculate the magnitude and direction of the force
between the cable and the wall and of the force between the steel beam and the wall.