Work and Kinetic Energy Lesson 7 by Owen Borville 11.22.2025
Work is the transfer of energy by a force acting on an object as it is displaced. The work W that a force F does on an object is the product of the magnitude F of the force, times the magnitude d of the displacement, times the cosine of the angle θ between them: W = Fdcos θ.
The SI unit for work and energy is the joule (J), where 1 J = 1 N * m = 1 kg * m^2/s^2.
The work done by a force is zero if the displacement is zero or if the displacement is perpendicular to the force. The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.
Net work (Wnet) is the work done by the net force acting on the object. Work done on an object transfers energy to the object. The translational kinetic energy of an object of mass m moving at speed v is KE = 1/2mv^2.
The work-energy theorem states that the net work Wnet on a system changes its kinetic energy: Wnet = 1/2mv^2-(1/2mv0^2)
Gravitational Potential Energy: Work done against gravity in lifting an object becomes potential energy of the object-Earth system. The change in gravitational potential energy, ΔPEg, is ΔPEg = mgh with h being the increase in height and g the acceleration due to gravity. The gravitational potential energy of an object near Earth's surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy ΔPEg, have physical significance. As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increased speed, so that ΔKE = - ΔPEg.
Conservative forces are forces in which work depends only on the starting and ending points of a motion, and not on the path taken. Conservative forces can have potential energy just as gravitational forces. The potential energy of a spring is PEs = 1/2kx^2, where k is the spring's force constant and x is the displacement from its undeformed position.
Mechanical energy = KE + PE for a conservative force. If there are only conservative forces acting in a system, the total mechanical energy is constant.
KE + PE = constant or KEi + PEi = KEf + PEf for conservative forces only, where i and f are initial and final values=(conservation of mechanical energy)
Nonconservative forces are forces which work depends on the path. Friction is an example of a nonconservative force that changes mechanical energy into thermal energy.
Work (Wnc) done by a nonconservative force changes the mechanical energy of a system: Wnc = ΔKE + ΔPE or KEi + PEi + Wnc = KEf + PEf.
When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton's Laws.
The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
When all forms of energy are considered, conservation of energy is = KEi + PEi + Wnc + OEi = KEf + PEf + OEf, where OE is all other forms of energy besides mechanical energy.
Common forms of energy are electrical energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
Energy is often used to do work, but it is not possible to convert all the energy of a system to work.
Eff (efficiency of a machine or human) = Eff = Wout/Ein, where Wout is useful work output and Ein is the energy consumed.
Power is the rate at which work is done, or average power P for work W done over a time t, P = W/t. The SI unit for power is the watt (W), where 1W = 1J/s. The power of many devices such as electric motors is also often expressed in horsepower (hp), where 1 hp = 746 W.
The indefinitely small increment of work done by a force acting over an indefinitely small displacement is the dot product of the force and the displacement. The work done by a force over a finite path is the integral of the infinitesimal increment of work done along the path.
dW = F * d r = |F||d r| cos θ
WAB (path A to B) = ∫F * dr
The work done against a force is the negative of the work done by the force.
Wfr (constant kinetic friction) = -fk |lAB|
The work done by a normal or frictional contact force must be determined in each particular case. The work done by the force of gravity, on an object near the surface of the Earth, depends only on the weight of the object and the difference in height through which it moved. The work done by a spring force, acting from an initial position to a final position, depends only on the spring constant and the squares of those positions.
Wgrav,AB = -mg (yB - yA)
Wspring,AB = -(1/2k)(xB^2 - xA^2)
The kinetic energy of a particle is the product of one-half of its mass and the square of its speed, for non-relativistic speeds. The kinetic energy of a system is the sum of the kinetic energies of all the particles in the system. Kinetic energy is relative to a frame of reference, is always positive, and is sometimes given unique names for different types of motion.
K = 1/2mv^2 = p^2/2m
Work-energy theorem: The net force on a particle is equal to its mass times the derivative of its velocity and the integral for the net work done on the particle is equal to the change in the particle's kinetic energy.
Wnet = KB - KA
Power is the rate of doing work, or the derivative of work with respect to time. During a time interval, work done is the integral of the power supplied over the time interval. The power delivered by a force, acting on a moving particle, is the dot product of the force and the particle's velocity.
P = dW/dt
P = F * v (dot product)
Work is the transfer of energy by a force acting on an object as it is displaced. The work W that a force F does on an object is the product of the magnitude F of the force, times the magnitude d of the displacement, times the cosine of the angle θ between them: W = Fdcos θ.
The SI unit for work and energy is the joule (J), where 1 J = 1 N * m = 1 kg * m^2/s^2.
The work done by a force is zero if the displacement is zero or if the displacement is perpendicular to the force. The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.
Net work (Wnet) is the work done by the net force acting on the object. Work done on an object transfers energy to the object. The translational kinetic energy of an object of mass m moving at speed v is KE = 1/2mv^2.
The work-energy theorem states that the net work Wnet on a system changes its kinetic energy: Wnet = 1/2mv^2-(1/2mv0^2)
Gravitational Potential Energy: Work done against gravity in lifting an object becomes potential energy of the object-Earth system. The change in gravitational potential energy, ΔPEg, is ΔPEg = mgh with h being the increase in height and g the acceleration due to gravity. The gravitational potential energy of an object near Earth's surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy ΔPEg, have physical significance. As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increased speed, so that ΔKE = - ΔPEg.
Conservative forces are forces in which work depends only on the starting and ending points of a motion, and not on the path taken. Conservative forces can have potential energy just as gravitational forces. The potential energy of a spring is PEs = 1/2kx^2, where k is the spring's force constant and x is the displacement from its undeformed position.
Mechanical energy = KE + PE for a conservative force. If there are only conservative forces acting in a system, the total mechanical energy is constant.
KE + PE = constant or KEi + PEi = KEf + PEf for conservative forces only, where i and f are initial and final values=(conservation of mechanical energy)
Nonconservative forces are forces which work depends on the path. Friction is an example of a nonconservative force that changes mechanical energy into thermal energy.
Work (Wnc) done by a nonconservative force changes the mechanical energy of a system: Wnc = ΔKE + ΔPE or KEi + PEi + Wnc = KEf + PEf.
When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton's Laws.
The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
When all forms of energy are considered, conservation of energy is = KEi + PEi + Wnc + OEi = KEf + PEf + OEf, where OE is all other forms of energy besides mechanical energy.
Common forms of energy are electrical energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
Energy is often used to do work, but it is not possible to convert all the energy of a system to work.
Eff (efficiency of a machine or human) = Eff = Wout/Ein, where Wout is useful work output and Ein is the energy consumed.
Power is the rate at which work is done, or average power P for work W done over a time t, P = W/t. The SI unit for power is the watt (W), where 1W = 1J/s. The power of many devices such as electric motors is also often expressed in horsepower (hp), where 1 hp = 746 W.
The indefinitely small increment of work done by a force acting over an indefinitely small displacement is the dot product of the force and the displacement. The work done by a force over a finite path is the integral of the infinitesimal increment of work done along the path.
dW = F * d r = |F||d r| cos θ
WAB (path A to B) = ∫F * dr
The work done against a force is the negative of the work done by the force.
Wfr (constant kinetic friction) = -fk |lAB|
The work done by a normal or frictional contact force must be determined in each particular case. The work done by the force of gravity, on an object near the surface of the Earth, depends only on the weight of the object and the difference in height through which it moved. The work done by a spring force, acting from an initial position to a final position, depends only on the spring constant and the squares of those positions.
Wgrav,AB = -mg (yB - yA)
Wspring,AB = -(1/2k)(xB^2 - xA^2)
The kinetic energy of a particle is the product of one-half of its mass and the square of its speed, for non-relativistic speeds. The kinetic energy of a system is the sum of the kinetic energies of all the particles in the system. Kinetic energy is relative to a frame of reference, is always positive, and is sometimes given unique names for different types of motion.
K = 1/2mv^2 = p^2/2m
Work-energy theorem: The net force on a particle is equal to its mass times the derivative of its velocity and the integral for the net work done on the particle is equal to the change in the particle's kinetic energy.
Wnet = KB - KA
Power is the rate of doing work, or the derivative of work with respect to time. During a time interval, work done is the integral of the power supplied over the time interval. The power delivered by a force, acting on a moving particle, is the dot product of the force and the particle's velocity.
P = dW/dt
P = F * v (dot product)