Roller Coaster
Solution:
1. This is a simple application of the conservation of energy law;
E_{Pi }+ E_{Ki} = E_{Pf} + E_{Kf}
Note that in the absence of friction, the mass term can cancel out of the equation as it is in every term,
h_{f} is considered zero elevation, or the reference point point and thus has the value of zero. The value of h_{i} is the sum of 65 m and 20 m, so the equation for v_{f} is given by
2. A free body diagram of the car at the bottom of the hill, where the radius of curvature for the track is 60 m, and the velocity is as calculated above yields the following equation.
3. This is once again a conservation of energy problem where h_{i} is considered zero elevation, or the reference point point, thus having the value of zero. The value of h_{f} is 60 m, so the equation for v_{f} is given by
4. When the car is in the banked turn it is the xcomponent of the normal force, F_{Nx}, that supplies the unbalanced force towards the center of the circle, and is therefore the centripetal force. This means that the ycomponent of the normal force, F_{Ny},_{ }and weight, W, must be balanced. Division of the resulting equations allows one of the unknowns, namely F_{N}, to be cancelled, leaving the variable theta.
5. This is once again a conservation of energy problem where h_{f } is considered zero elevation at the bottom of the loop, thus having the value of zero, however at the top of the loop h_{f} is 30 m. The value of h_{i} is the sum of 60 m and 10 m.
The equation for v_{f} at the bottom of the loop is given by,
The equation for v_{f} at the top of the loop is given by,
6. A free body diagram of the car at the top of the loop, where the radius of the loop is 15 m, and the velocity is as calculated above yields the following equation.
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