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Week 6 - Rigid Body Motion I [5] [7]
Summary
We will cover the motion of rigid bodies using the Lagrangian.
Reading List
- ``Lagrangian Dynamics''
REF: Wells, D. A. 1967, Chapter 8.
- ``Classical Mechanics''
REF: Goldstein, H. 1980, Chapters 4-5.
- ``Mechanics''
REF: Landau, L. D., Lifshitz, E. M. 1988, Chapter 6.
Problem Set - Optional Problems Answers
Problem 1 - Marble and Bowl
A marble of mass M and radius r rolls without slipping on the
inside surface of a spherical bowl of radius R. You can assume that
angular velocity of the marble at any moment is parallel to the
surface of the bowl (ω &cdot R=0) where R
is a radial vector between the centre of the spherical bowl and the
point where the marble makes contact with the bowl). The only
external force is gravity.
What is the moment of inertia of the marble about an axis passing
through its centre of mass (assume that the density of the marble is
uniform)? What is the total kinetic energy of the marble as it moves
along the surface of the bowl?
What is the Lagrangian for the marble? What quantities are conserved
during the particle's motion? Write out the equations of motion for
the marble. What is the frequency of small oscillations about the
point of equilibrium? What does the particle's trajectory look like
for these small oscsillations?
Problem 2 - Back to UPS
The UPS depot at YVR has a series of conveyor belts. Sometimes to aid
in sorting, a package is transferred from one belt to another. A
metal bar drops to impede the motion of the package along the first
belt, and a piston pushes the package onto a second belt going
perpendicular to the first belt. Both belts travel at the same
velocity and have the same coefficient of kinetic friction (dry
friction). You may assume that the frictional force between the belt
and the box is proportional to the fraction of the box on the belt.
Write out the equation of motion for the box (assume the box is
rectangular and aligned in the direction of the motion of the belts).
If you neglect the acceleration of the box, you can solve for the
velocity of the box as a function of the fraction of the box on each
belt. Integrate up the velocity to find the time as a function of position.
Problem 3 - Battle Bot
A four-wheeled remote control vehicle is fitted with a large counterrotating drum.
The angular velocity of the drum is horizontal and perpendicular to
the direction of the vehicle's motion. The top of the drum moves so
that is velocity is opposite to the forward velocity of the vehicle.
Explain why steering the vehicle is unstable when the vehicle is going
forward and statble when the vehicle is going backward.
Last modified: Wednesday, 30 November 2005 12:14:26
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