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# 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, Ep, for conservative force is defined by relation

where:

W is work done by the force

Epi  and Epf 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

Eki, E pi are initial kinetic and potential energies respectively

Ekf, 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

xo is length of unstretched spring

xi and xf 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 vi and vf 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

Total kinetic energy of moving object

where:

vc is linear speed of mass center of the object

is angular speed of the object

m is mass of the object

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