Inertia, Force, and Momentum

Here are the updated study notes:

Inertia and its Types

• Inertia is the inherent property of a body that resists changes to its state of rest or uniform motion.
• Inertia represents an object's inability to change its current state, whether at rest or in motion.
• Inertia is directly related to mass; a greater mass implies a greater inertia.
• There are three types of inertia:
  • Inertia of rest: Inability to change the state of being at rest.
  • Inertia of motion: Inability to change the state of motion.
  • Inertia of direction: Inability to change direction.

Force and its Effects

• Force is what alters an object's state of motion, rest, shape, or direction.
• Forces can start or stop motion, change an object's direction, and alter its shape.

Momentum Explained

• Momentum is the product of an object's mass and velocity (p = mv).
• Momentum is related to the impact an object can create.
• An object with higher mass or velocity will have a higher momentum and create a bigger impact.

Newton's First Law of Motion

• An object at rest will stay at rest, and an object in motion will stay in motion with the same speed and direction unless acted upon by an external force.
• The net external force is the cause of acceleration that is non-zero.
• A body continues to be in a state of rest or uniform motion in a straight line unless compelled to change that state by some external force.

Newton's Second Law of Motion

• The rate of change of linear momentum is directly proportional to the force applied to the body.
• This change takes place in the direction of the applied force.
• Key terms in the definition: rate of change of momentum, force, and direction.
• The rate of change of momentum is directly proportional to the applied force and occurs in the direction of the force.
• Force is the rate of change of momentum: F = dp/dt
• F = ma is the more common (but incomplete) simplified form of Newton's Second Law, valid only when mass is constant.
• When mass is variable, the formula f=d(mv)/dt must be used.
• When mass is constant F = m dv/dt

Applying Newton's Laws: Examples & Problem Solving

• Given the rate of dust falling onto a moving train, calculate the extra force required to maintain constant velocity using the formula f=v(dm/dt)
• Finding force needed to stop an object.
• In problem-solving, first list the 'given' parameters, then what is 'to find', then the relevant formula, and finally the solution.
• Including the correct unit is important.

Newton's Third Law of Motion

• For every action, there is an equal and opposite reaction.

Frames of Reference: Inertial vs. Non-Inertial

• Inertial frame of reference: A frame where an object is at rest or moving at a constant velocity (acceleration = 0).
• Non-inertial frame of reference: A frame that is accelerating.
• Newton's laws are obeyed in inertial frames.
• In non-inertial frames, Newton's laws are not directly applicable and require the consideration of pseudo forces.
• A pseudo force is an apparent force that arises in a non-inertial frame of reference due to the acceleration of the frame itself.
• A pseudo force 'takes' or uses the acceleration of the system, has direction opposite to system's acceleration, F = -ma.

Fundamental Forces in Nature

• There are four fundamental forces in nature: Gravitational, Electromagnetic, Strong Nuclear, and Weak Nuclear.
• Gravitational Force:
  • Always attractive, and the weakest force.
  • Has infinite range.
  • Depends on the masses of the objects and the distance between them.
• Electromagnetic Force:
  • Can be attractive or repulsive.
  • Stronger than the gravitational force, and has a long range
  • responsible for most of the forces we encounter.
• Strong Nuclear Force:
  • The strongest force.
  • Binds protons and neutrons together inside the nucleus of an atom.
  • Short range.
• Weak Nuclear Force:
  • Responsible for radioactive beta decay.
  • Short range.
  • Not as weak as gravitational force.

Contact vs. Non-Contact Forces

• Contact forces require physical contact between objects.
• Non-contact forces do not require physical contact.

Real vs. Pseudo Forces

• Real forces arise from the interaction between objects.
• Pseudo forces arise in non-inertial frames of reference due to the acceleration of the frame.

Lift Problems and Apparent Weight

• In a stationary or constant velocity lift, the apparent weight is equal to the actual weight (mg).
• In an accelerating lift, the apparent weight changes, and calculations must account for net acceleration.
• In a lift with a broken cable that falls in freefall the tension is zero.

Conservative and Non-Conservative Forces

• Conservative forces: Work done is independent of path.
  • Mechanical energy is conserved.
• Non-conservative forces: Work done depends on the path.
  • Mechanical energy is not conserved. Examples are air resistance, friction.

Work Done Explained

• Work Done = Force x Displacement (W = F ∙ s)
• if force is constant, the is W = Fs cos(θ)
• if the force varies with respect to distance, then Work is derived from: W = ∫ F(x) dx
• Work is equal to the area under the curve on a force vs. displacement graph.
• Work Done = Change in Kinetic Energy (Work-Energy Theorem)
• Work Done = Change in Potential Energy
• Work done equals zero when there is no displacement, no force, or the force is perpendicular to the displacement (θ = 90°).

Kinetic & Potential Energy

• Kinetic energy: Energy possessed by a body due to its motion.
• Potential energy: Energy possessed by a body due to its position. There are various potential energies to be aware of.

Law of Conservation of Energy

• Energy cannot be created nor destroyed, it only changes form
• If there is a decrease in potential energy, then that will be converted into kinetic energy.
• Kinetic energy is 1/2 m v^2
• Potential energy is shown as U generally, may be gravitational e.g. m g h.
• Total energy remains constant at all points in a closed system.
• You may have an a potential to work problem where energy is lost to air resistance.

Conservation of Momentum

• Rocket goes up, and gas goes down, all adding up the external forces remain zero.
• In initial condition the bullet is still and the total momentum is zero, the firing adds momentum in one direction. Therefore to ensure that total sum remains zero there is an opposite recoil.
• Recoil velocity is velocity that the gun moves back to balance momentum.

Different Types of Forces

• Weight is a gravity that acts on mass, and is always pulling downwards.
• Normal is what a surface pushes back always at 90 degrees to the pushing force.
• Tension will always appear on a string, always in string direction, and pulls.
• Springs resist compression or extension that pushes things to go back to zero/rest.
• Friction, always resist motion, always surface applied.

Free Body Diagrams (FBDs)

• FBD is the visual breakdown of forces, must know each force's direction / source for accurate FBD

Pulleys, Ramps, Blocks

• These tools always need a separate FBD.
• At the end always sum to zero.
• Look at driving and opposing forces.
• May need to equate forces.
• Constant velocity or at rest, means constant
• Inclined ramps will need sine to resolve components of forces.

Collisions

• Collision always involves strong short lived force.
• Elastic means kinetic energy and momentum conserved.
• Inelastic means kinetic energy no preserved.
• If two items stick together then that is also inelastic.

Coefficient of Restitution

• v2-v1 / sqrt(1u^2^ - u1^2^)
• 1 means fully elastic
• Less than 1 means plastic / inelastic
• Zero means fully plastic / inelastic

Two Dimensional Collisions and Impact

• Tangent is established when two objects collide. The center point is also known as line of impact.
• Can now look at components of incoming forces u1 and u2 , along with v1 and v2.

Heights and Bounces

• hN = e^2^N * h1 (formula for each subsequent height.

Impulses

• Impact over time
• J = Force * dt
• Force times change in momentum
• In graphic terms the area under the Force Time Graph.

Torque

• The spin or moment that is equal to rate.
• r x Force and Torque is r * Force.

Static Equilibrium

• All forces and torques are set to zero so there is rest.

Potential Energies at Rest

• Stable lowest potential energy and always reverts to stable state
• Unstable it will always seek a new position
• Neutral won't be altered in new position.