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2D Dynamics
With 2D dynamics, we can explain the orbit of the planets around the Sun, the grandfather clock, and the perfect angle to throw a snowball to nail your nemesis as they run away from you.
Take gravitational acceleration as \( g. \)
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Old time mills for grinding wheat or corn into flour were often powered by a water wheel. Some water wheels had water falling over paddles, other water wheels dipped their paddles in a moving river, which as it flowed past turned the wheel. One particular mill has a water wheel with radius \(3~\mbox{m}\) suspended above a river such that the bottom edge of the paddles are just in contact with and move with the river water. If the river flows at \(0.5~\mbox{m/s}\) then what is the period of the spinning water wheel?
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A roller coaster works by gravitational energy. The coaster car is pulled up to a high point and then released, rolling downwards on the track through all manners of curves and loops. I have a short roller coaster car that I pull up to the top of a hill of height \(H\). The coaster car is released from this height and must go around a perfectly circular vertical loop with a radius of 20 meters (and the bottom of the loop is on the ground). If I don't want the coaster car to fall off the loop at any point, what should be the minimum value of \(H\) in meters?
Details and assumptions
- Neglect friction and air resistance.
- Treat the roller coaster car as a point.
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What is the maximum speed with which a \( 1600 \text{ kg} \) car can make a left turn around a curve of radius \( 49 \text{ m} \) on a level (unbanked) road without sliding?
The static friction constant between the car and the road is \( 0.1, \) and the gravitational acceleration is \( g = 10 \text{ m/s}^2. \)
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