Problem 2 Loaded Hoop on a Conveyor Belt
Deadline March, 31 2012
A loaded hoop consists of a 'load' or point mass m, embedded in a rigid hoop with mass
M and radius R. The hoop is on a flat horizontal conveyor belt. At time t, the load subtends
an angle θ with the vertical while the horizontal and vertical coordinates of the center C of
the hoop are X and Y respectively.
In this problem assume that the hoop does not tilt and is always in the same vertical
plane. The gravitational acceleration is g.
Use the following conventions for the derivatives of any function f : f˙ ≡ df/dt
and f' ≡ df/dθ
y
M
g = acceleration
due to gravity
C(X,Y)
Y=R θ Conveyor Belt, velocity = v(t)
m
x
X
Part 1. Preliminary Calculations
For Part 1, assume that the hoop is always in contact with the conveyor belt and that X
and θ are independent (i.e. do not assume any constraint such as a non-slip condition etc.).
1.A. Acceleration: Express the horizontal and vertical components, ax and ay, of the acceleration of the load in the laboratory frame in terms of R, X, θ, θ˙, and θ¨.
1.B. Energy: Use the convention that the gravitational potential energy of the loaded hoop
is zero when the load is at the bottom. Express the total mechanical energy, E, of the
loaded hoop in terms of M, m, g, R, θ, θ˙, and X˙.
Part 2. Oscillation on a Stationary Conveyor Belt
For Part 2 assume that the conveyor belt is at rest. If the loaded hoop is given a small
angular displacement from the vertical and released from rest the system will oscillate.
Problem 2 Loaded Hoop on a Conveyor Belt Page 1 of 3
Problem 2 Loaded Hoop on a Conveyor Belt
Deadline March, 31 2012
2.A. Period when friction is negligible: Assuming that friction is negligible, determine the
period of the small oscillation, TA of the loaded hoop.
2.B. Period when the hoop does not slip: Assuming that friction is high enough to ensure
that there is no sliding, determine the period of small oscillation, TB of the loaded
hoop.
Part 3 Stable Angular Orientations on an Accelerating Conveyor Belt
For Part 3, assume the following:
• The loaded hoop is placed on the conveyor belt at time t = 0 and initially θ = X˙ =
X = 0.
• The velocity at any point of the belt is given by v = Kt where K is a proportionality
constant.
If the loaded hoop is placed on the conveyor belt at t = 0 with carefully chosen initial values
of θ, then it will maintain a stable angular orientation provided K/g, μs, μk, and m/M
satisfy appropriate conditions – assume that these conditions are satisfied only in Part 3.
3.A. Determine the stable orientation angle θ = α as a function of m/M when m/M ≥ 1/7,
μk = √3/3, K = √3g and μs = 1.0.
3.B. Determine the stable orientation angle θ = α as a function of m/M when m/M ≥ 1/7,
μk = √3/3, K = √3g and μs = 2.0.
Part 4. Oscillation, Rotation, and Sliding on an Accelerating Conveyor Belt
For Part 4, assume the following:
• M = m
• The loaded hoop is placed on the conveyor belt at time t = 0 and initially θ = X˙ =
θ˙ = X = 0.
• The velocity at any point of the belt is given by v = Kt where K is a positive constant.
• Let ω ≡ dθ/dt.
4.A. When the coefficient of static friction is extremely high (as will be assumed here), two
types of motion may ensue-
Type A: The loaded hoop undergoes angular oscillations forever and never slides.
Type B: The loaded hoop completes at least one rotation without sliding.
Problem 2 Loaded Hoop on a Conveyor Belt Page 2 of 3
Problem 2 Loaded Hoop on a Conveyor Belt
Deadline March, 31 2012
(4.A.1) Type A Motion. For Part 4.A.1, suppose that the loaded hoop is observed to
undergo Type A Motion and maximal angular displacement of the load is observed
to be θmaximum ≡ β = 30°. Deduce an algebraic expression for K/g in terms of β
and determine its numerical value.
(4.A.1) Type B Motion. When K/g exceeds a threshold value γ, the loaded hoop undergoes Type B motion. Determine the numerical value of γ to two significant
figures.
4.B. Sliding Motion. Now assume that the value of K/g is the same as the value that you
were asked to calculate in Part 4.A.1, but this time do not assume that the coefficient
of static friction is extremely high. When the coefficient of static friction is less than a
threshold value μs,0, the loaded hoop will not undergo Type A motion because it will
slide. Determine the numerical value of μs,0 to two significant figures.
Hint:
1. d²θ/dt² = ω dω/dθ
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