Classical Mechanics

Mechanics Warmups

Mechanics Warmups: Level 4 Challenges

         

A dynamical system is described by the equation r˙=f(r)\dot{r} = f(r) where f(r)f(r) is an nthn^\text{th} order polynomial with roots {r1,,rn}\{r_1,\ldots,r_n\}. In other words, the roots of ff are the steady states of the system (r˙(ri)=0\dot{r}(r_i) = 0).

Suppose the system is placed into one of the steady states rir_i and is perturbed very slightly away to ri+Δrr_i+\Delta r. How does the perturbation change over time?

Three objects of same heat capacity with temperature T1=200 KT_1=200~\mbox{K}, T2=400 KT_2=400~\mbox{K} and T3=400 KT_3=400~\mbox{K} exchange heat with each other. They are isolated from the rest of the universe. Find the highest possible temperature one of them can reach in kelvin.

Details and assumptions:

  • You may connect them to a heat engine and/or to a refrigerator
  • The objects do not collide to generate heat energy

Find the Time Period of oscillations (in sec\displaystyle \text{sec}) for the arrangement as shown in the figure.

Details and Assumptions:

  • The pulleys are smooth and massless.
  • k1=100N/m\displaystyle k_1 = 100 N/m
  • k2=200N/m\displaystyle k_2 = 200 N/m
  • M=4kg\displaystyle M = 4 kg
  • g=9.8m/s2\displaystyle g = 9.8 m/s^2

A dynamical system is described by the equation r˙=f(r)\dot{r} = f(r) where f(r)f(r) is an nthn^\text{th} order polynomial with roots {r1,,rn}\{r_1,\ldots,r_n\}. In other words, the roots of ff are the steady states of the system (r˙(ri)=0\dot{r}(r_i) = 0).

Suppose the system is placed into one of the steady states rir_i and is perturbed very slightly away to ri+Δrr_i+\Delta r. How does the perturbation change over time?

Toy helicopters with rechargeable batteries fly for a few minutes on a single charge. Manufacturers want to choose the right size of battery to achieve the longest flight time between charges. A larger battery stores more energy, but also increases the mass of the helicopter so it takes more energy to keep it in the air. Our question is, for a helicopter and battery type that behave as specified in the assumptions below, what is the linear size ll of the battery in mm that will maximize the time the helicopter can hover in place?

Details and assumptions

  • The acceleration of gravity is 9.8 m/s2-9.8~\mbox{m/s}^2.
  • The base mass m0m_0 of the helicopter without the battery is 50 g50~\mbox{g}.
  • The battery is a uniform cube with side length ll and density ρ=4000 kg/m3\rho= 4000 ~\mbox{kg/m}^3.
  • The total energy contained in the battery does not quite scale as the volume does, but instead scales as l2.7l^{2.7}.
  • The emitted voltage from the battery is constant and does not change with ll. All that changes is how long the battery lasts. Therefore the rotors spin and push air in the same way no matter what the battery size is.
  • Assume the mass of the air pushed by the rotors is a constant value, independent of rotor speed. In an actual helicopter the mass increases with speed, effectively from Bernoulli's equation, but that's more complication than we want to go into.
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