AP1- U3_WEP (1)
Unit 3 Work, Energy & Power
Study Chapter 7
Objectives
Describe the work done on an object or system by a given force or collection of forces.
Describe the translational kinetic energy of an object in terms of the object’s mass and velocity.
Describe the potential energy of a system.
Describe the energies present in a system.
Describe the behavior of a system using conservation of mechanical energy principles.
Discuss how the selection of a system determines whether the energy of that system changes.
Work
Definition
Work occurs when an external force causes displacement.
Formula: W = F d cos(θ) where F is constant and motion aligns with the direction of the force.
Unit: Joule (1 J = 1 N m)
Type: Scalar quantity (can be positive, negative or zero).
Work Quantities
Maximum Positive Work:
Occurs when force is parallel to displacement: W = F d cos(0°) = F d
Work at an Angle
Only the component of the force parallel to the displacement performs work.
No Work Condition
When force is perpendicular to motion (θ = 90°):
W = F d cos(90°) = 0
Maximum Negative Work
When the force opposes the displacement (θ = 180°):
W = F d cos(180°) = -F d
Work and Force Graph
The area under a force vs position graph indicates the work done:
For F(x) from xa to xb: Area = Work done.
Net Work
Net Work (Wnet): Sum of work done by all acting forces on an object.
Formula: Wnet = Fnet Δx
Summary of Work
No work is done by a force if it does not cause displacement.
Forces perpendicular to displacement do no work (e.g., normal force and gravitational force).
Wnet: Wnet = Fnet Δx
Positive and Negative Work
Positive Work:
When force and displacement are in the same direction or when the force causes an increase in speed.
Negative Work:
When force and displacement are in opposite directions (e.g., friction) or when the force decreases speed.
Kinetic Energy (K)
Definition: Energy due to the motion of an object.
Formula: K = ½ m v²
Unit: Joules (always positive scalar).
The Work-Energy Theorem
Theorem: Net work done by external forces equals the change in kinetic energy.
Formula: Wnet = ΔK = Kf - Ki
If Wnet > 0, KE increases
If Wnet < 0, KE decreases
If Wnet = 0, KE remains unchanged (ΔKE = 0).
Potential Energy (U)
Definition
Energy associated with the position of objects within a system.
Property of the system, not just the object.
Type: Scalar, can be negative depending on position.
Gravitational Potential Energy (Ug)
Measured relative to a zero level (lowest point in scenario).
Defined to be zero when objects are infinitely far away; becomes negative as objects approach.
Change in Ug
Formula near the surface of a planet where gravitational field (g) is constant:
ΔUg = mgΔy
where m = mass, g = 9.8 m/s², y = position above zero level.
Work done by Gravity
Wg = -ΔUg = -mg(yf - yi)
Elastic Potential Energy
Definition
Energy stored in materials that are elastic and flexible, depending on the amount compressed or stretched.
For a spring
Formula: Us = ½ k(Δx)²
where Δx is distance compressed or stretched from equilibrium, k is the spring constant.
Formula for work done by the spring: Ws = -ΔUs.
Energy Due to Conservative Forces
Work done by conservative forces is path independent and zero along a closed path.
Change in potential energy (ΔU) can be recovered (e.g., gravity, elastic forces).
Energy Due to Non-Conservative Forces
Work done is path dependent and not zero along a closed path.
Can dissipate mechanical energy as thermal energy or sound.
Example: kinetic friction, drag, and propulsive forces.
Energizes Present in a System
A single object only possesses kinetic energy.
Systems with interacting objects via conservative forces or reversible shape changes (e.g., springs) have both kinetic and potential energies.
Mechanical Energy
Definition: Total mechanical energy of a system = sum of its kinetic and potential energies.
Formula: Emech = K + U
Conservation of Mechanical Energy Principles
Any change in a type of energy must be balanced by an equivalent change in other energy types or energy transfer with surroundings.
Total energy of an isolated system is constant unless work is done from outside.
System Selection and Conservation
Energy is conserved in all interactions.
If work done on the system is zero and no non-conservative interactions occur, total mechanical energy remains constant.
Nonzero work indicates energy transfer between system and environment.
Conservation of Mechanical Energy with No Friction
System must be closed (no mass added or removed) and isolated (no external forces).
Assumes Wnet = 0, resulting in: Ui + Ki = Uf + Kf
Conservation of Energy
Total energy formula: E = U + K + Eth = Constant
Energy changes accounted for via: ΔE = ΔU + ΔK + ΔEth = 0
Motion with Gravity
Conservation equations: MEi = MEf
K i + U gi = K f + U gf
Motion with Gravity and Friction
Equation: Wnet = ΔU + ΔK + ΔEth = 0
Breakdown: Ugi + Ki = Ugf + Kf + ΔEth.
Conservation of Horizontal Spring Energy
Equation: ½ kx12 + ½ mv12 = ½ kx22 + ½ mv22
Relates mechanical energy in spring oscillation.
Pendulum Energy
Equation: mgh = ½mv_max²
Relates potential and kinetic energy throughout maximum and minimum points of swing.
Spring and Pendulum Energy Profile
Visual representation of total energy, potential energy, and kinetic energy.
Energy Bar Charts
Example: A ball tossed into the air shows that the sum of kinetic and gravitational potential energy (K + U_G) remains constant.
Power
Definition: Rate at which energy is transferred or converted with respect to time.
Average Power: P_avg = ΔE/Δt = W_net/Δt
Instantaneous Power: P_inst = Fv = Fv cos(θ)
Unit: Watts (W = J/s)