Work, Energy, and Momentum: Study Notes for College Physics

Work and Energy

Definition of Work

Work is a measure of energy transfer when a force acts upon an object to move it. The work done by a force depends on both the magnitude of the force and the displacement of the object in the direction of the force.

• Work can be positive, negative, or zero depending on the direction of force relative to displacement.

• Formula: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• Dot Product: The dot product of force and displacement yields a scalar value for work.

• Example: If a box is pulled up an incline, multiple forces (gravity, tension, friction, normal force) may do work, but only those with a component along the displacement contribute non-zero work.

Path Independence and Conservative Forces

For certain forces, the work done is independent of the path taken. These are called conservative forces.

• Examples: Gravity, spring force, electric force.

• Non-conservative force: Kinetic friction (work depends on path).

• Work by gravity: The total work done by gravity depends only on the initial and final positions, not the path taken.

• Example: Moving a mass around a loop and returning to the same point results in zero net work by gravity.

Work Done by Springs

Spring forces are conservative. The force exerted by an ideal spring is proportional to its displacement from equilibrium.

• Hooke's Law: $F = -kx$

• Spring constant (k): Measures stiffness; units are N/m.

• Work required to stretch a spring: $W = \frac{1}{2}kx^2$

• Example: Compressing or stretching a spring stores potential energy.

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

Kinetic energy is the energy of motion. It is always positive or zero and is a scalar quantity.

• Formula: $KE = \frac{1}{2}mv^2$

• Example: If two cars have the same mass but different speeds, the faster car has much more kinetic energy (proportional to the square of speed).

Work-Energy Theorem

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

• Formula: $W_{net} = \Delta KE = KE_{final} - KE_{initial}$

• Example: A catcher stopping a baseball does negative work, reducing the ball's kinetic energy.

Potential Energy and Conservation of Energy

Potential Energy

Potential energy is stored energy due to position or configuration. Common forms include gravitational and elastic (spring) potential energy.

• Gravitational Potential Energy: $PE_{grav} = mgy$

• Elastic Potential Energy: $PE_{spring} = \frac{1}{2}kx^2$

• Potential energy can be negative depending on the choice of reference point.

Conservation of Energy

If only conservative forces act, the total mechanical energy (kinetic + potential) of a system remains constant.

• Formula: $E_{total} = KE + PE$

• Example: A block sliding down a frictionless ramp converts potential energy to kinetic energy.

• With friction: Some mechanical energy is converted to heat, so $E_{total}$ decreases.

Energy Graphs

Graphs of potential energy versus position help visualize energy changes in systems like roller coasters.

• At turning points: Kinetic energy is zero, and all energy is potential.

• At other points: $KE = E_{total} - PE$

Power

Definition of Power

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

• Formula: $P = \frac{W}{t}$

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

• Example: Climbing stairs quickly requires more power than climbing slowly.

Momentum and Impulse

Definition of Momentum

Momentum is the product of mass and velocity and is a vector quantity.

• Formula: $\vec{p} = m\vec{v}$

• Units: kg·m/s

• For a system: Total momentum is the sum of individual momenta.

Impulse

Impulse is the change in momentum resulting from a force applied over a time interval.

• Formula: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$

• Impulse is the area under the force vs. time curve.

Conservation of Momentum

Principle of Conservation

In an isolated system, the total momentum remains constant if no external forces act.

• Internal forces cancel out due to Newton's Third Law.

• Collisions: Momentum is conserved in all collisions, but kinetic energy is only conserved in elastic collisions.

• Types of collisions:

  • Elastic: Both momentum and kinetic energy are conserved.

  • Inelastic: Momentum is conserved, but kinetic energy is not.

  • Perfectly inelastic: Objects stick together after collision.

Momentum in Two Dimensions

Momentum conservation applies in each direction independently. In 2D collisions, vector addition is used to analyze momentum before and after interaction.

• Example: Ball 1 strikes Ball 2; the final momentum components must satisfy conservation laws.

Summary Table: Types of Collisions

Type	Momentum Conserved?	Kinetic Energy Conserved?	Result
Elastic	Yes	Yes	Objects bounce, no energy loss
Inelastic	Yes	No	Objects may deform, some energy lost
Perfectly Inelastic	Yes	No	Objects stick together, maximum energy loss

Additional info: These notes expand on brief points from the original materials, providing full academic context, definitions, and examples for clarity and completeness.