Classical Mechanics

Potential Energy

Potential Energy: Level 2-3 Challenges

         

Lukla airport in Nepal is one of the strangest in the world. Built to support tourism to the Himalayas, the airport has a single landing runway. What is more, the runway is only 20 m wide, 450 m long, and a 2,800 m cliff at the runway's end, leaving little room for error. In fact, the airport can only be used by so-called Short Takeoff and Landing planes (STOL). Helping somewhat is a 12% incline in the runway from start to finish, so that planes rise through over the course of their deceleration.

Suppose a STOL plane's landing speed is 45 m/s (\approx 100 mph). Neglecting any other effects like wind flaps, or drag, how small will the plane's velocity (in m/s) be at the top of the runway?

Details

  • The runway itself is 450 m long, i.e. if you walked from the bottom to the top, you'd walk 450 m along the runway.
  • An ff% incline means that if you walk a distance dd along an incline, your rise is given by fd/100fd/100.

Climbing out of bed is sometimes hard in the morning. If you have a mass of 60 kg60~\mbox{kg}, what is the minimum amount of work in joules you have to do to get out of bed? Assume your center of mass when lying in your bed is 0.75 m0.75~\mbox{m} above the floor, and your center of mass when standing is 1.25 m1.25~\mbox{m} above the floor.

Details and assumptions

  • The acceleration due to gravity is 9.8 m/s2-9.8~\mbox{m/s}^2.

Consider a system of NN particles whose coordinates are r={rix,riy,riz}\mathbf{r} = \{r_i^x, r_i^y, r_i^z\}, and whose velocities are {r˙i}\{\dot{\mathbf{r}}_i\}.

Each pair of particles (i,j)\left(i, j\right) interacts through a potential V(ri,rj)V\left(\mathbf{r}_i, \mathbf{r}_j\right) which has no direct dependence on time. As the system evolves in time, which of the following bulk quantities must be conserved?

A particle is constricted to move along the positive xx-axis under the influence of a potential energy:

U(x)=3x+7xU(x) = \frac{3}{x} + 7x

Find the point of equilibrium for the particle. (Round your answer to three decimal places.)

Consider a system with NN particles whose coordinates are ri={rix,riy,riz}\mathbf{r}_i = \{r_i^x,r_i^y,r_i^z\}, and whose velocities are r˙i\dot{\mathbf{r}}_i. The energy of the system is described by the kinetic energy 12mr˙2\frac{1}{2}m\sum \dot{\mathbf{r}}^2 and an effective potential energy term V(r1,,rN).V\left(\mathbf{r}_1, \ldots, \mathbf{r}_N\right).

All we know about VV is that it depends on the positions only through their differences r1r2\mathbf{r}_1 - \mathbf{r}_2, i.e. V({ri},t)=V(r1r2,r1r3,,t).V(\{\mathbf{r}^i\}, t) = V\left(\mathbf{r}_1-\mathbf{r}_2, \mathbf{r}_1-\mathbf{r}_3, \ldots, t\right). As this system evolves in time, which of the following bulk quantities must be conserved?

Assumptions

  • All interactions of the system with the outside world are described by V.V.
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