chapter7_class_slides
Goals for Chapter 7
Overview energy.
Study work as defined in physics.
Relate work to kinetic energy.
Consider work done by a variable force.
Study potential energy.
Understand energy conservation.
Include time and the relationship of work to power.
Introduction
Previous chapters focused on motion using Newton's three laws.
Understanding forces and motion can be complex.
Introduction of energy as the next concept to understand.
An Overview of Energy
Energy is conserved: it is a fundamental concept.
Kinetic Energy (KE):
Describes motion and relates to mass and speed squared.
Formula: ( KE = \frac{1}{2} mv^2 )
Units: joules (kg m²/s²).
Thermal Energy: Kinetic energy on a microscopic level.
Potential Energy:
Associated with forces such as gravity, elastic forces, electromagnetic forces, and nuclear forces.
Includes energy carried by waves (e.g., light energy from the sun).
Chemical Energy: Arises from electromagnetism; released through metabolism.
Nuclear Energy: Results from strong and weak nuclear forces.
Conservation of energy principles underlie phenomena from everyday life to advanced scientific concepts like nuclear fusion.
Potential Energy (1 of 2)
Conservative forces (e.g., gravity, elastic forces) involve stored energy due to spatial arrangements.
Gravitational Potential Energy (U_grav):
Near the earth's surface, it can be expressed as:
( U_{grav} = mgy )
How Money is Like Energy
Money exists in multiple forms (coins, bills, checks, bank accounts).
Transformations of money (e.g., cashing a check) do not change its total amount.
Money can be transferred to alter the total amount in one form.
Money can be stored or spent similar to how energy is stored or used.
What is "Work" as Defined in Physics?
Work is defined as the product of a constant force (F) acting through a parallel displacement (s).
Mathematically: ( W = F \cdot s ) (dot product).
Only the component of force parallel to displacement contributes to work.
Work can be:
Positive (energy is added),
Negative (energy is taken from an object),
Zero (no change in energy).
The Work-Energy Theorem
The work-energy theorem connects work to kinetic energy and the conservation of energy principles.
Total work (W_total) can be positive, negative, or zero.
If ( W_{total} = 0 ), kinetic energy does not change.
Potential energy should be considered accordingly.
Relationship: ( W = \Delta KE ) (change in kinetic energy).
Gravitational Potential Energy
Change in gravitational potential energy relates to work done by gravity.
Positive work done by gravity decreases potential energy.
Example with positive and negative work:
When moving downward, potential energy decreases;
When moving upward, potential energy increases.
Conservative and Nonconservative Forces
Conservative Forces: Allow conversion between kinetic and potential energy (e.g., gravity, springs).
Work done is reversible and independent of the path taken.
Work is zero if starting and ending points are identical.
Nonconservative Forces:
Do not conserve mechanical energy, e.g., friction.
Potential Energy (2 of 2)
Change in potential energy from conservative forces is tied to net work:
( W = -\Delta U_{grav} = -U_{f} + U_{i} )
Total mechanical energy conservation principle connects kinetic and potential energy.
A Solved Baseball Problem – Example
Initial speed when a ball is thrown upwards results in decreased kinetic energy as it converts to potential energy.
At the peak, kinetic energy is zero, and potential energy is maximum.
As the ball falls, potential energy converts back to kinetic energy.
Applications of Force and Resultant Work
Positive work: Force has a component in the direction of displacement.
Negative work: Force has a component opposing displacement, slowing the object down.
Perpendicular forces do no work on the object.
Work Done by Several Forces – Example
Involves multiple forces including weight and normal forces.
Net work calculated by summing individual works of all forces acting on the object.
Work Done by a Varying Force (1 of 2)
Work done by a changing force can be integrated over its path.
Total work = Area under the force vs. position curve.
Work Done by a Varying Force (2 of 2)
Example: Spring force follows Hooke's law:
( F_{spring} = kx )
Work done is equal to the area of the triangle created in force-position graph.
Work Done on a Spring Scale – Example
Compressed springs in scales store energy, which can be calculated from examples.
Energy Stored in Spring Displacement
Spring's elastic potential energy described by:
( U_{el} = \frac{1}{2} kx^2 )
Potential Energy on an Air Track with Mass and Spring
Employs conservation of energy principles to determine mechanical energy states at different positions.
Conversion and Conservation
Conservation of mechanical energy principle combines kinetic and potential energy conversions, enhancing systems' dynamics.
Mechanical Energy Calculation – Example
Determines total mechanical energy using mass, heights, and spring energy components.
Conservative and Nonconservative Forces
Total mechanical energy is conserved only under conservative forces.
Nonconservative forces, like friction, require the inclusion of work done in calculations.
Non-conservative force of Friction Generates Internal Energy
Mechanical energy converts into internal or thermal energy through microscopic vibrations in solid structures.
Nonconservative Forces – Example
Includes work done by an external constant force along with spring mechanics.
Conservative Forces Overview
Work independent of path for conservative forces; if endpoints are the same, total work equals zero.
Power – Consider Work and the Time to Do It (1 of 2)
Average power calculated as work done over time: ( P_{av} = \frac{\Delta W}{\Delta t} )
Units: 1 watt = 1 joule/second (J/s).
Power – Consider Work and the Time to Do It (2 of 2)
Example: Work done during a marathon stair climb relates power output to physical effort, calculated using weight and height.