MEEN 363/501 Dynamics & Vibrations, Fall 2022
HW 1
Objectives: describe particle motion with different coordinate systems and reference points,
practice coordinate transformations.
Note:
- Convert all numbers to SI units (meters, seconds, radians, etc) for final answers.
- Show your work (required for full credit), and circle final answers.
- Crunching numbers and generating plots in the software of your choice (python, excel,
matlab, wolframAlpha, etc) is encouraged; however, you still need to explain your work
and reasoning.
- Be careful about signs!
Problem 1 (5 points)
A paddle steamer is moving down a river at a speed of 20 mph, with its paddle wheel churning at
25 rpm. The wheel has a radius r = 3 meters. At time t=0, the paddles (labeled (a-h) are in the
configuration shown in the figure above.
At time t=0:
a. Which is the fastest moving paddle relative to a person standing on shore?
- Report the magnitudes of the velocity and acceleration of that paddle relative to
the shore observer.
b. Which is the slowest moving paddle relative to a person standing on shore?
- Report the magnitudes of the velocity and acceleration of that paddle relative to
the shore observer. .
c. A seagull flying by at a constant speed of 40 mph and an angle of β = 30° off the water’s
surface:
- What is the velocity of paddle b in the fixed, rotated coordinate system ijbird?
- Draw a diagram showing the velocity vectors for paddle b and the bird, and the
unit vector directions for both the cartesian system (ij) and rotated coordinate
system (ijbird).
- What is the velocity of paddle b relative to the bird?
Problem 2 (5 points)
You are skiing down the slope of Mt. Aggie. Your path is described by equation:
𝑋
𝑌(𝑋) = 𝐴𝑐𝑜𝑠( 𝑑 π) + 𝐴
where d = 1000 and A = 300. Gravity (g) = 9. 8
𝑚
2
𝑠
Velocity (tangential velocity) along the arc is simply the potential energy converted to kinetic energy:
𝑣𝑡(𝑋) = 2𝑔ℎ
where ℎ = 2𝐴 − 𝑌(𝑋)
Tasks:
For the interval x E (0,d):
a. Plot skier velocity vt(x) vs x. Does it match your intuition of the skier’s speed?
b. Find the equation and plot the radius of curvature ρ(x) vs. x
c. Find the equations and plot the skier’s tangential(at), normal (an), & total (||a||) acceleration vs. x
For the location x = d/2:
d. Find skier’s velocity [vx; vy] and acceleration [ax; ay] in cartesian coordinates
e. Find skier’s velocity [vr; vθ] and acceleration [ar; aθ] in polar coordinates
Discuss:
f. What physical, operation and design considerations have we neglected in this example, and how
would they affect the skier’s motion? Think of shape, friction, etc. Explain in words.