Forces, Newton's Laws, Momentum, and Energy: Mini-Textbook Study Notes

Forces and Interactions

Interaction and Force Pairs

In physics, an interaction occurs between two objects and always involves a pair of forces acting on the two objects. These forces are called action-reaction pairs.

• Action force: The force exerted by object A on object B.

• Reaction force: The force exerted by object B on object A.

• These forces are equal in magnitude and opposite in direction, and always act on different objects.

• Example: When you push on a wall, the wall pushes back on you with an equal and opposite force.

Newton's Third Law of Motion

Statement and Implications

Newton's Third Law of Motion states: "Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first."

• Action and reaction forces are co-pairs of a single interaction.

• Neither force exists without the other.

• They always act on different objects.

• Example: The tires of a car push back against the road, and the road pushes the tires forward.

Action and Reaction on Different Masses

• The same force exerted on a small mass produces a large acceleration; on a large mass, a small acceleration (per Newton's Second Law).

• Example: When a cannon is fired, the cannonball (small mass) accelerates much more than the cannon (large mass).

Applications and Examples

• When stepping off a curb, Earth pulls you downward, and you pull Earth upward with an equal force. However, Earth's acceleration is imperceptible due to its large mass.

• In collisions (e.g., a bus and a bug), both experience forces of equal magnitude, but the effects differ due to their masses.

• Two people of equal mass pushing off each other on ice will move away at equal speeds in opposite directions.

System Definition and Internal vs. External Forces

• Forces internal to a system (e.g., an apple and orange pulling on each other) cancel and do not cause acceleration of the system.

• External forces (e.g., friction from the floor) are required to accelerate the system as a whole.

Flight and Lift

• Helicopters and birds fly by pushing air downward; the air pushes them upward (lift).

• Airplane wings deflect air downward to produce lift.

• A heavier glider must push air downward with greater force to maintain altitude.

Vectors and Components

Vector Components

A vector can be resolved into perpendicular components, typically vertical and horizontal.

• Resolution is the process of determining these components.

• Example: Pulling a sled at an angle, the force can be split into horizontal (forward motion) and vertical (lifting) components.

Forces on Inclined Planes

• On an incline, the weight (mg) acts vertically downward and remains constant.

• The normal force (N) decreases as the angle increases.

• At 90°, the net force on the block is mg.

Summary of Newton's Three Laws of Motion

• First Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion at constant speed in a straight line unless acted upon by a net force.

• Second Law (Acceleration): The acceleration of an object is directly proportional to the net force and inversely proportional to its mass: $a = \frac{F_{\text{net}}}{m}$

• Third Law (Action-Reaction): For every action, there is an equal and opposite reaction.

Momentum

Definition and Formula

Momentum is a property of moving objects, defined as the product of mass and velocity:

• Formula: $p = mv$

• Momentum is a vector quantity (has direction).

• Example: A moving boulder has more momentum than a stone at the same speed; a faster boulder has more momentum than a slower one of the same mass.

Impulse

Impulse is the product of force and the time interval over which it acts. It equals the change in momentum:

• Formula: $J = F \Delta t$

• Impulse changes momentum: $F \Delta t = \Delta p$

• Increasing the time of contact reduces the average force for the same change in momentum (e.g., landing on a soft surface).

Bouncing and Impulse

• Impulses are greater when objects bounce, as the momentum changes direction, requiring a larger impulse.

• Example: Catching and throwing back a ball requires more impulse than just stopping it.

Conservation of Momentum

Law of Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant:

• Formula: $p_{\text{before}} = p_{\text{after}}$

• Example: When a cannon fires, the momentum gained by the cannonball is equal and opposite to the momentum gained by the recoiling cannon.

Collisions

• Elastic collision: Objects rebound without lasting deformation or heat generation.

• Inelastic collision: Objects deform and/or generate heat.

• In all collisions (without external forces), total momentum is conserved.

• Example: Two identical freight cars coupling together after collision move at half the initial speed of the moving car.

Collisions in Two Dimensions

• When objects collide at angles, use vector addition (parallelogram method) to find the resultant momentum.

• Example: The vector sum of the momenta of pieces after a firecracker explodes equals the initial momentum.

Energy

Definition and Forms

Energy is the ability of a system to do work. It is a conserved quantity and can be transferred or transformed but not created or destroyed.

• Forms include mechanical, thermal, chemical, nuclear, and more.

• Matter is the substance that has mass and occupies space.

Work

• Work is done when a force acts on an object and the object moves in the direction of the force.

• Formula: $W = Fd$

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

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

Power

• Power is the rate at which work is done.

• Formula: $P = \frac{W}{t}$

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

• Example: Running up stairs quickly requires more power than walking up slowly, even if the work done is the same.

Mechanical Energy

• Mechanical energy is due to position (potential energy) or motion (kinetic energy), or both.

Potential Energy

• Potential energy (PE): Stored energy due to position or configuration.

• Gravitational potential energy: $PE = mgh$ (where $h$ is height above a reference point)

• Example: Water in an elevated reservoir, a stretched bow.

Kinetic Energy

• Kinetic energy (KE): Energy of motion.

• Formula: $KE = \frac{1}{2}mv^2$

• If speed doubles, kinetic energy quadruples.

• Example: A moving car has both momentum and kinetic energy.

Work-Energy Theorem

• Any change in kinetic energy is the result of work done: $W = \Delta KE$

• Example: The work done by brakes to stop a car equals the loss in kinetic energy.

Conservation of Energy

• Energy cannot be created or destroyed, only transformed from one form to another.

• Example: In a bow and arrow, potential energy is converted to kinetic energy and some heat.

Kinetic Energy and Momentum Compared

• Both are properties of moving objects.

• Momentum is a vector (directional), kinetic energy is a scalar (no direction).

• Momentum depends linearly on velocity; kinetic energy depends on the square of velocity.

Machines and Efficiency

Machines

• Machines multiply forces or change the direction of forces but cannot create energy.

• Lever: Rotates on a fulcrum; allows a small force over a large distance to move a large load a short distance.

• Pulley: Changes the direction of the input force; systems of pulleys can multiply force.

Efficiency

• Efficiency is the percentage of work input converted to useful work output.

• Formula: $\text{Efficiency} = \frac{\text{useful work output}}{\text{total work input}} \times 100\%$

• Example: A machine that is 30% efficient converts 30% of input energy to useful work; the rest is lost, usually as heat.

Recycled Energy

• Reusing energy that would otherwise be wasted (e.g., using waste heat from power plants to heat buildings).

Energy for Life

• The human body requires energy, obtained from food (hydrocarbons) that react with oxygen to release energy.

Sources of Energy

Solar and Renewable Energy

• The Sun is the primary source of most energy on Earth (solar, wind, hydroelectric).

• Photovoltaic cells convert sunlight directly into electricity.

• More solar energy reaches Earth in one hour than all human energy consumption in a year.

Fuel Cells

• Fuel cells generate electricity by combining hydrogen and oxygen, producing water and electric current.

Nuclear and Geothermal Energy

• Nuclear energy is stored in uranium and plutonium; geothermal energy comes from underground heat.

• Dry-rock geothermal power uses water injected into hot rock to produce steam for turbines.

Quantity	Definition	Formula	Unit
Momentum	Inertia in motion	$p = mv$	kg·m/s
Impulse	Change in momentum	$J = F \Delta t$	N·s
Work	Force times distance	$W = Fd$	Joule (J)
Power	Rate of doing work	$P = \frac{W}{t}$	Watt (W)
Kinetic Energy	Energy of motion	$KE = \frac{1}{2}mv^2$	Joule (J)
Potential Energy	Energy due to position	$PE = mgh$	Joule (J)
Efficiency	Useful work output / total work input	$\text{Efficiency} = \frac{\text{useful work output}}{\text{total work input}} \times 100\%$	Percent (%)

Additional info: Some explanations and examples have been expanded for clarity and completeness, and standard equations have been included for reference.