WORK The work dW done on a particle displaced along differential path dr, by an object exerting force F is defined as A B F dr The SI unit of work is 1J = 1N·1m We can define work in an integral form: For the work done on an object, one must specify on which point associated with the object the work is done (the center of mass, the point of the force application... ).

kinetic energy A particle with mass m, moving with speed v has kinetic energy K of Note. Kinetic energy is used to describe the motion of an object. When an object is approximated by a particle, the kinetic energy defined above is called the translational kinetic energy of the object.

Work-Energy Theorem I In an inertial reference frame, the work dW done by all the forces exerted on the particle (the net force) is equal to the change in the kinetic energy dK of the particle dW = dK work & energy In an integral form:  W =  K

Power The power of a force is defined as the rate at which work is done by that force. The SI unit of work is 1W = 1J/1s inverse relation: relation to force:

Conservative Interaction If the work done by a “force” on an object moving between two positions is independent of the path of the motion, the force is called a conservative force. A B All other forces are nonconservative. (Theorem) The work done by a conservative force around a loop (the object returns to its initial position) is zero. conservative interactions : gravitational, elastic, electrostatic.

Potential Energy If a force exerted on a particle is conservative, the change in potential energy dU from one position to another is defined by the work dW performed by that force dU  - dW (or  U = -  W ) This definition assigns potential energy only with accuracy to a constant.

Gravitational Potential Energy m W h The gravitational potential energy U g of a particle with mass m, placed at a position with a vertical component different by h from the reference location is U g = mgh h UgUg dr

Elastic Potential Energy The elastic potential energy that an ideal spring has by virtue of being stretched or compressed is x UsUs bungee

Gravitational Potential Energy M m r dr F Where choose the reference?  The gravitational potential energy of a particle with mass m, placed at distance r from another particle with mass M is What about the reference at the surface?

Mechanical Energy The sum of the kinetic and the potential energy of a particle is called the total mechanical energy of the particle. E  K + U motion related energy position related energy

Energy Conservation All concepts of energy are defined in such a way that energy can neither be created nor destroyed, but can be converted from one form to another. The total energy of an isolated system is always constant.

Work-Energy Theorem II If some forces exerted on a particle are conservative, the work W nc, done by all forces not included in the potential energy, is equal to the change in the mechanical energy E of the particle. bungee

Potential Energy and the Force The conservative force is opposite to the gradient of the potential energy caused by this force because F dr and x y z

Example. Gravitational potential energy at the surface m W = - mg z x y z

Example. Gravitational potential energy x y z F r

Example. What should be the initial speed of an object which is supposed to escape the gravitational field of the Earth? M m v probe v Earth in the reference frame of the Earth: conservation of the mechanical energy in the reference frame of the object: What is wrong