Physics 141Mechanics Lecture 10 Force and PotentialYongli Gao A conservative force can be obtained from the derivative of its potential. From the definition of the potential energy, or, Example: spring

Potential and Force in 3-D In 3-D, the potential energy is where is called partial derivative with respect to x, and in taking the derivative assuming y and z are constants. We can have the gradient operator defined as

Potential Energy Curve We’ll see the relationships between kinetic energy, potential energy, force, turning point, etc., from a potential energy curve to be drawn in the lecture.

Energy and Reference Frame The work done by a force may depend on the inertial reference frame of the observation. An airline stewardess does a little work pushing a cart observed by the passengers, but much more by the ground crew. What you need to realize is that the change of kinetic energy also depends on the reference frame and W=  K precisely holds in each frame. Using Galilean transformation, y x Frame A Frame B x’ y’ r BA r PA r PB V AB

We see that the difference observed in two inertial frames about work done and kinetic energy change are exactly the same.

Example: Stewardess and Carts The stewardess push a cart of mass m=50 kg forward with a force F=20 N for a distance d=5.0 m in time  t=5.0 s in an airplane moving at speed V=180 m/s (648 km/hr), what will be the work done by her seen by the passengers and by the ground control tower? What will be the kinetic energy change viewed by the two parties if no friction? Solution: It is a 1-D problem. Observed by the passengers,

Observed by the control tower, We see that although the work done and kinetic energy observed in different frames are different, but the law relates work and kinetic energy, is observed in each of the frames.

*Mass and Kinetic Energy From Einstein’s theory of special relativity, the energy of any object is where m 0 is called the rest mass, the mass measured in the reference frame the object is at rest, and v its velocity. The kinetic energy is the difference between the energy and the rest energy Using Taylor expansion, we have

Example: Relativistic Energy The power consumption of the US is 5.0 x 10 9 W. How long will a 50 kg object last if all its energy is released? Solution: The total energy relativistically is E=m 0 c 2 =50 kg x (3.0 x 10 8 m/s) 2 = 4.5 x J The time it’ll last t=E/P= 4.5 x J/(5.0 x 10 9 W) = 9.0 x 10 8 s ≈30 yr

*Energy Quantization Whenever there is a confinement, the energy will be quantized. An atom is an example: the energy levels of electrons confined in an atom are quantized. For a hydrogen atom, the allowed energy levels are Quantum # (n)Energy (eV) … nE 1 /n 2. … ∞ 0

Center of Mass We have so far been treating objects as particles. That is, we ignore that shape and size of the object. We’ll see that for any object of finite size and shape, there is in general one special point at which all we learned about the motion of a particle is precisely accurate. That point is called center of mass. For a system of n particles, the center of mass is defined as

Example: Center of Mass of Particles Suppose that two particles are along the x-axis, m 1 at x 1 and m 2 at x 2, and m 2 =2m 1, find the position of the center of mass. Solution: