Work and Energy

SI units & Physics constants

Work and energy investigates different forms of energy, their transmutation, and interrelationship between energy and work

Here (all units see here):

m is mass of object

 is force vector

 is displacement vector

 is angle between vectors  and

 is velocity

General formulas

Work done by constant force during small displacement is defined by scalar product

Work done by force during motion between initial  and final  positions is defined by definite integral over trajectory

Instantaneous power is defined as derivative with respect to time

Average power required to perform work W during time t 

Relation between power, force and velocity

Kinetic energy of object

Conservative force is the force when the work it does on object is independent of the path between the object's initial and final positions. Otherwise the force is called nonconservative

Potential energy, E_p, for conservative force is defined by relation

where:

W is work done by the force

E_pi  and E_pf are potential energies at initial and final positions respectively

Gravitational potential energy of object at height h

where g is free-fall acceleration

Total mechanical energy

Work-Energy Theorem

General Work-Energy theorem

where:

W is net work done by external nonconservative forces acting on the system

E_ki, E _pi are initial kinetic and potential energies respectively

E_kf, E _pf are final kinetic and potential energies respectively

Principal of conservation of energy for conservative system (without nonconservative forces)

Work and energy of spring

Work done by spring

where:

k is spring constant

x_o is length of unstretched spring

x_i and x_f are initial and final lengths of the spring respectively

Potential energy of spring stretched to length x

Energy conservation principal for spring-mass system

where v_i and v_f are initial and final speeds of the mass respectively

Work and energy of rotational motion

Work done by torque

where:

 is component of net torque parallel to axis of rotation

 and  are initial and final angular positions of object

Kinetic energy of rotating object

where:

I is moment of inertia of object about its axis of rotation

 is angular speed

Total kinetic energy of moving object

where:

v_c is linear speed of mass center of the object

 is angular speed of the object

m is mass of the object

I_c is moment of inertia of the object about the axis of rotation passing through its center of mass