Work and Kinetic Energy

Work

The transfer of energy that occurs when a force causes a displacement.

Work Symbol

W

SI Unit of Work

joule (J)

One Joule

The work done when a force of one newton moves an object one metre.

Dimensions of Work

kg·m²/s²

Conditions for Work

A force must act on an object and the object must undergo displacement.

Positive Work

Occurs when the force and displacement are in the same direction.

Negative Work

Occurs when the force and displacement are in opposite directions.

Zero Work

Occurs when there is no displacement or when the force is perpendicular to the displacement.

Constant Force

A force whose magnitude and direction remain unchanged during displacement.

Work Done by a Constant Force

The product of the force and the displacement in the direction of the force.

Work Formula

W = Fd cos θ

Force Symbol

F

Displacement Symbol

d

Angle Symbol

θ

Meaning of θ

The angle between the force vector and the displacement vector.

Maximum Work

Occurs when θ = 0°.

Maximum Work Formula

W = Fd

Minimum Work

Occurs when θ = 180°.

Minimum Work Formula

W = −Fd

Zero Work Condition

Occurs when θ = 90°.

Dot Product

A mathematical operation that multiplies two vectors to produce a scalar.

Dot Product Formula

A · B = AB cos θ

Work as a Dot Product

W = F · d

Scalar Quantity

A quantity having magnitude only.

Vector Quantity

A quantity having both magnitude and direction.

Work

A scalar quantity.

Applied Force

A force exerted directly on an object.

Displacement

The change in position of an object.

Component of Force Along Displacement

F cos θ

Perpendicular Component of Force

F sin θ

Effective Force

The component of force parallel to the displacement.

Parallel Force Component

The portion of the applied force that performs work.

Perpendicular Force Component

Does not contribute to work.

Work Done by Gravity

The work performed by the gravitational force during vertical displacement.

Work Done by Gravity Formula

Wg = mg(yi − yf)

Positive Gravitational Work

Occurs when an object moves downward.

Negative Gravitational Work

Occurs when an object moves upward.

Work Done by the Normal Force

Usually zero because the normal force is perpendicular to displacement.

Work Done by Friction

Usually negative because friction opposes motion.

Net Work

The sum of the work done by all forces acting on an object.

Net Work Symbol

Wnet

Work Depends On

Force, displacement, and the angle between them.

Mechanical Work

Work associated with forces acting through a displacement.

Variable Force

A force whose magnitude or direction changes during displacement.

Work Done by a Variable Force

The work equal to the area under the force-displacement graph.

Work Done by a Variable Force Formula

W = ∫F·dr

Integral of Work

The summation of infinitesimal amounts of work over a displacement.

Differential Work

dW = F·dr

Force-Displacement Graph

A graph showing force as a function of displacement.

Area Under a Force-Displacement Graph

Represents the work done by the force.

Positive Area on an F-x Graph

Represents positive work.

Negative Area on an F-x Graph

Represents negative work.

Net Work

The algebraic sum of the work done by all forces acting on an object.

Net Work Formula

Wnet = ΣW

Work-Energy Theorem

The net work done on an object equals the change in its kinetic energy.

Work-Energy Theorem Formula

Wnet = ΔKE

Change in Kinetic Energy Formula

ΔKE = KEf − KEi

Kinetic Energy

The energy possessed by an object because of its motion.

Kinetic Energy Symbol

KE

Kinetic Energy Formula

KE = ½mv²

SI Unit of Kinetic Energy

joule (J)

Initial Kinetic Energy

KEi = ½mvi²

Final Kinetic Energy

KEf = ½mvf²

Change in Kinetic Energy

The difference between the final and initial kinetic energies.

Positive Net Work

Increases the kinetic energy of an object.

Negative Net Work

Decreases the kinetic energy of an object.

Zero Net Work

Produces no change in kinetic energy.

Relationship Between Work and Kinetic Energy

Work transfers energy to or from an object.

Relationship Between Speed and Kinetic Energy

Kinetic energy is proportional to the square of the speed.

Effect of Doubling Speed

Kinetic energy becomes four times greater.

Effect of Tripling Speed

Kinetic energy becomes nine times greater.

Effect of Doubling Mass

Kinetic energy doubles if speed remains constant.

Moving Object

Possesses kinetic energy.

Stationary Object

Has zero kinetic energy.

Energy

A scalar quantity measured in joules.

Work

A method of transferring energy.

Mechanical Energy

The sum of kinetic and potential energy.

Net Work by Multiple Forces

The sum of the work done by each individual force.

Force Parallel to Motion

Produces maximum work.

Force Opposite Motion

Produces negative work.

Force Perpendicular to Motion

Produces zero work.

Conservative Force

A force whose work depends only on the initial and final positions.

Nonconservative Force

A force whose work depends on the path taken.

Examples of Conservative Forces

Gravitational force and spring force.

Examples of Nonconservative Forces

Friction and air resistance.

Mechanical Energy

Can be transferred through work.

Energy Transfer

Occurs when work is performed on or by an object.

Work Done on an Object

Changes the object's kinetic energy.

Work Done by the Net Force

Equals the change in kinetic energy.

Principle of Work and Energy

The net work performed on an object changes its kinetic energy.

Spring Force

The restoring force exerted by a stretched or compressed spring.

Spring Force Formula

F = −kx

Hooke's Law

The restoring force of a spring is directly proportional to its displacement from equilibrium.

Hooke's Law Constant

The constant that measures the stiffness of a spring.

Hooke's Law Constant Symbol

k

SI Unit of Spring Constant

newton per metre (N/m)

Spring Displacement

The distance the spring is stretched or compressed from equilibrium.

Spring Displacement Symbol

x

Restoring Force

A force that acts opposite the displacement and tends to restore equilibrium.