module 4 lecture 8: Momentum, Energy, Work, and Simple Machines

Momentum and Collisions

Definition and Properties of Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction.

• Formula: $p = m v$

• Unit: kilogram meter per second (kg·m/s)

• Properties: Momentum depends on both mass and velocity. An object with greater mass or higher velocity has more momentum.

• Example: A moving car carries momentum, not force. The momentum is the product of its mass and velocity.

Law of Conservation of Linear Momentum

The law of conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant before and after a collision.

• Mathematical Statement: $\text{Total momentum before collision} = \text{Total momentum after collision}$

• Equation: $\sum m v_{\text{before}} = \sum m v_{\text{after}}$

• Application: This law applies to all types of collisions (elastic and inelastic) as long as no external force acts on the system.

Impulse and Change in Momentum

Impulse is the product of force and the time interval over which it acts. It is equal to the change in momentum of an object.

• Formula: $\Delta p = F \Delta t$

• Unit: Newton-second (N·s) or kg·m/s

• Concept: The force of impact can be increased or decreased by adjusting the time over which the momentum changes.

• Example: In a car-bug collision, both the car and the bug experience equal and opposite forces, impulses, and changes in momentum, as per Newton's third law.

Energy and Work

Definition of Energy

Energy is the capacity to do work. It is a conserved quantity in physics, meaning the total energy in a closed system remains constant, though it can be transformed or transferred.

• Quote (Feynman): Energy is a numerical quantity that does not change when nature undergoes changes. This is the law of conservation of energy.

• Forms of Energy: Mechanical, thermal, chemical, electrical, etc.

Work in Physics

Work is done when a force causes displacement in the direction of the force. It is a measure of energy transfer.

• Formula: $W = F d$ (when force and displacement are in the same direction)

• Unit: Joule (J), where $1 \text{ J} = 1 \text{ N} \cdot 1 \text{ m}$

• Mechanical Work: The term was coined by Gaspard-Gustave Coriolis. It refers to the energy transferred by a force.

• Example: Lifting a barbell twice as heavy over the same distance requires twice as much work.

Key Questions and Examples

• Work against a stationary wall: No work is done on the wall if it does not move, regardless of the force applied.

• Work in lifting: Lifting a heavier object or lifting the same object a greater distance requires more work.

• Work with constant force: Pushing a cart twice as far with the same force does twice as much work.

Forms of Mechanical Energy

Kinetic Energy

Kinetic energy is the energy of motion. It depends on the mass and the square of the velocity of an object.

• Formula: $KE = \frac{1}{2} m v^2$

• Properties: Kinetic energy is a scalar quantity and always positive. Doubling the speed of an object quadruples its kinetic energy.

Potential Energy

Potential energy is stored energy due to an object's position or configuration. The most common form is gravitational potential energy.

• Gravitational Potential Energy Formula: $PE = m g h$

• Examples: A stretched bow, a compressed spring, or an object held at a height all possess potential energy.

• Work-Energy Relationship: The work done to lift an object is stored as gravitational potential energy.

Mechanical Energy Conservation

Mechanical energy is the sum of kinetic and potential energy in a system. In the absence of non-conservative forces (like friction), mechanical energy is conserved.

• Formula: $E_{\text{mechanical}} = KE + PE$

• Conservation: $KE_{\text{initial}} + PE_{\text{initial}} = KE_{\text{final}} + PE_{\text{final}}$

Kinetic Energy and Momentum Compared

• Momentum: Vector quantity ($p = m v$), can be positive or negative depending on direction.

• Kinetic Energy: Scalar quantity ($KE = \frac{1}{2} m v^2$), always positive.

• Relationship: For a given kinetic energy, a lighter object must move faster than a heavier one. For two objects with the same kinetic energy, the one with greater mass has greater momentum.

Work-Kinetic Energy Theorem

The work-kinetic energy theorem states that the net work done on an object is equal to its change in kinetic energy.

• Formula: $W = KE_f - KE_i = \Delta KE$

• Application: Doubling the speed of an object requires four times the work, since kinetic energy depends on the square of velocity.

• Example: The stopping distance of a car increases with the square of its speed. If speed doubles, stopping distance quadruples.

Power

Power is the rate at which work is done or energy is transferred.

• Formula: $P = \frac{W}{t}$

• Unit: Watt (W), where $1 \text{ W} = 1 \text{ J/s}$

Simple Machines

Definition and Types

Simple machines are devices that provide a mechanical advantage, allowing the same work to be done with a different balance of force and distance.

• Ramp (Inclined Plane): Allows lifting a load with less force over a greater distance. The work done is independent of the ramp's steepness.

• Lever: A rigid bar that pivots about a fulcrum. Applying a smaller force over a larger distance can lift a heavier load.

• Pulley: A wheel with a rope that changes the direction of the applied force and can provide mechanical advantage. Multiple pulleys can reduce the force needed to lift a weight.

Machine	Mechanical Advantage	Example
Ramp	Less force, more distance	Loading a cart onto a truck
Lever	Small force, large distance	Seesaw, crowbar
Pulley	Force divided by number of ropes	Flagpole, crane

Energy Sources

Energy sources are essential for powering society. They are classified as renewable or nonrenewable.

• Renewable: Can be replenished within a human lifetime (e.g., solar, wind, water, biomass, geothermal).

• Nonrenewable: Cannot be replenished on a human timescale (e.g., fossil fuels, nuclear fuel).

Summary of Key Formulas

• Force x distance (Work): $W = F d$

• Gravitational potential energy: $PE = m g h$

• Kinetic energy: $KE = \frac{1}{2} m v^2$

• Work-energy theorem: $W = \Delta KE$

• Law of conservation of energy: $E_{\text{total, initial}} = E_{\text{total, final}}$

Additional Examples and Applications

• Wind Turbines: When wind speed doubles, the power produced increases eightfold because both the mass flow rate and kinetic energy per unit mass increase.

• Comparing Balls: For two balls with the same kinetic energy, the lighter ball moves faster, but the heavier ball has greater momentum.

Additional info: Some explanations and context have been expanded for clarity and completeness, including the summary table of simple machines and the detailed comparison of kinetic energy and momentum.