Assignment 3 – Advanced Dynamics
Due: Sunday, 20 October 2019 at 11:00:00 pm

QUESTION 1: [60 MARKS]
A disk of mass m and radius R is rolling without slipping on a flat surface that is inclined by a constant
angle relative to the horizontal plane (figure 1). The plane of the disk can be inclined with respect
to the plane where it is rolling.
1. [10 pts] Find the equations of motion for the system (use a set of generalised coordinates of
your choice, along with any necessary constraint equations – be very clear about your choice
of generalised coordinates so that they are defined without ambiguity).
2. [50 pts] Choose non-zero numerical values for , , and and non-trivial initial
conditions for the system (that satisfy all constraint equations). State your choice of
parameters and initial conditions on your assignment. Simulate (using numerical
integration) and animate the system using MATLAB for some period of time, creating a
video of the animation in mp4 or avi format. Submit your animation video and MATLAB code
(.m file) along with the rest of the assignment. In your animation, the disc should not be
uniformly coloured, so that it is easy to see it rolling. The inclined surface should also be
displayed.

Figure 1

QUESTION 2: [40 MARKS]
The sphere in Figure 2 is rolling without slip on a flat surface. The system has 3 degrees of freedom.
The sphere has a uniformly-distributed mass and its radius is .

Figure 2
1. [20 pts] Find the equations of motion for the system (use a set of generalised coordinates of
your choice, along with any necessary constraint equations – be very clear about your choice
of generalised coordinates so that they are defined without ambiguity).
2. [20 pts] Assume that the sphere was at rest when an impulsive force hit it. The magnitude of
the impulsive force is � and its direction is 0 = �cossin0 �, where is a constant angle and
frame {0} is given in Figure 3. You can assume (for convenience) that the vector from the
impact point to the centre of mass of the sphere is parallel to axis 0.
Assume that the friction forces are not impulsive, and slip can occur during the impact.
Formulate the equations necessary to find the state of the sphere immediately after the
impact.

Figure 3