Motion in One Dimension: Kinematics

Motion in One Dimension

• Mastery of motion is critical for survival across biological processes like blood flow to satellite pursuits

Key Concepts

• Dynamics is the study of motion relating force and mass.
• Kinematics, a subset of dynamics, focuses on describing motion irrespective of its cause.
• Discussions will focus on kinematics, specifically regarding displacement, velocity, and acceleration.

Displacement

• Describes the change in an object's position in time and space.
• Describing motion needs a coordinate system and a specified origin.
• Frame of reference defines the starting point so is essential for solving mechanics problems.

Displacement Defined

• Displacement (∆x) is defined as the change in position of an object. Expressed as: ∆x = xf - xi
• A positive or negative answer indicates the displacement direction along the x-axis.
• Distance is the magnitude of displacement, while displacement is the change in position.

Velocity

• It is a vector quantity defined by both magnitude and direction.
• Average speed of an object over time is the path length divided by elapsed time.
• Average Speed = Path Length / Elapsed time
• Path length is the actual total length an object covers between two points.

Average Velocity

• Average velocity: a vector quantity possessing magnitude and direction
• Average velocity of an object in one dimension: can be positive or negative
• Depends on the sign of the displacement, with time always considered positive.
• Objects moving between two points via different paths share the same average velocity if their path length are different

Example: Football Run

• A player runs downfield from their own goal line to inches from a touchdown, reverses direction, and is tackled at the original starting point.
• The run lasted 25 seconds.
• Path length travelled: 200 yards.
• Displacement: 0 yards.
• Average velocity in the x-direction: 0 m/s.
• Average speed: 8 yards/sec.

Graphical Interpretation of Velocity

• Positions on a graph are a function of time elapsed since motion started.
• Average velocity of an object during time interval ∆t is equal to the slope of a straight line joining the initial and final points on a position vs. time graph.

Average Acceleration

• Describes the change of an object's velocity with time.
• Average acceleration (a) during a time interval ∆t is: a = (vf - vi) / (tf - ti) = ∆v / ∆t

Acceleration Example

• A car's velocity changes from -10 m/s to -20 m/s in 2 seconds along the negative x-direction
• Average acceleration is: ā = (-10 m/s - (-20 m/s)) / 2 s = -5 m/s²
• Negative acceleration indicates direction, not deceleration.

Motion Diagrams

• Illustrate movement patterns
• The distance between the car images can either stay consistent, increase, or decrease over time.
• The motion of the car demonstrates various states of constant/changing speed and acceleration.

Forces

• Commonly seen as a push or a pull on some object
• Commonly applied by humans in everyday life.
• Known fundamental forces: all field forces.
• Strong nuclear force between subatomic particles
• Electromagnetic forces between electric charges
• Weak nuclear force arises in certain radioactive decay processes
• Gravitational force acts between objects

Weight

• Weight (W): the downward force of gravity.
• It is a common force in our lives.
• Weight acts on objects that are stationary, moving horizontally, or moving vertically.

Friction

• Friction: resistance force to relative motion between bodies or substances in physical contact.
• Two types of friction: static and kinetic.
  • Static friction: occurs when there is no relative motion between two objects.

Example: Brakes

• Brakes: useful application of kinetic friction.
• Bicycle brakes: kinetic friction is used when pads are rubbed against the rim of a wheel.
• When brakes are applied: kinetic friction between pads and the rim/disk slows down a vehicle.

Newton's First Law

• An object remains at rest or maintains uniform motion with constant velocity unless acted upon by a net external force.
• Net force: vector sum of all external forces exerted on the object.
• Equal forces in opposite directions lead to a net force equal to zero.

Centripetal Force

• A force is required to keep objects following a curved path.
• Without centripetal force: an object will move at the same speed in a straight line.
• Formulated as: Fc = mv²/r
• This principle keeps spinning room amusement rides operating.

Example: Car

• 1000 kg car going at 10 m/s around a curve with a 20 m radius.
• Centripetal force is: F = (1000 kg * (10 m/s)^2) / 20 m = 5000 N.

More on Newton's First Law

• Net force is the vector sum of two applied forces in different directions when sailors push a boat.
• External force: must act on an object to adjust its speed, slow down, or change its direction of motion.

Mass

• Mass measures an object's resistance to acceleration.
• The effect of a net force on an object depends on the object's mass.
• With a very large mass, acceleration of the object will be small.
• Conversely, with a small mass, acceleration will be large.

Mass vs. Weight

• Mass: intrinsic property of matter, not dependent on external phenomena.
• Weight: the force arising from gravitational interaction.
• A hammer's mass will stays constant whereas weight varies based on location.

Newton's Second Law of Motion

• An object accelerates when a net external force acts on it.
• Net force equals the object's mass times acceleration: F = ma
• The unit of force is the newton (1 N = kg*m/s²).
• The acceleration of an object is proportional to the net force.
• Tripling the force on an object results in triple its acceleration

Example

• An airplane with mass of 2,000 kilograms is accelerating at 4 m/s².
• Applying F = ma, net force is: F = 2000 kg * 4 m/s² = 8000 N.

Example: Car Manufacturer

• An auto manufacturer builds a car accelerating uniformly from 0 to 27 m/s in 10 seconds.
• Given: mass = 1000 kg
• Solved as: F = (1000 kg)*(27 m/s - 0) / 10 = 2700 N

Newton's Third Law of Motion

• Forces come in pairs.
• If one object exerts a force on another, the second object exerts an equal and oppositely directed force on the first.

More on Newton's Third Law

• If object 1 and object 2 interact, then force F12 by object 1 on object 2 is equal in magnitude/opposite in direction to force F21 by object 2 on object 1.
• Forces in nature exist in pairs, and action/reaction forces act on different objects.

Action-Reaction Pairs

• Locomotion depends on action and reaction forces.
• In helicopters, the blades are arranged in 2 sets to maintain a stable position (demonstrates Newton’s Third Law).
• Earth exerts gravitational force (Fg) on any object. The reaction force is called normal force (-Fg). Related as: Fg = -Fg and n = -n.

More on Action-Reaction Pairs

• Calculate acceleration (a) of the man
• Determine the reaction force on the woman:
  • F = -R = -85N
• Calculate the woman's acceleration (aw):
  • aw = -F / m = -85.0 N / 55.0 kg = 1.55 m/s²

Law of Universal Gravitation

• Every object exerts a gravitational pull on every other object.
• The force is proportional to the masses of each object, inversely proportional to the square of the distance between centers.
• F = G(m1m2/d²), where G = 6.67 x 10-11, and d is distance between centers.

Gravitational Force on Earth and Beyond

• Gravitational force Earth exerts on other objects is inversely proportional to the square of distance from Earth's center.
• The Moon's centripetal acceleration is much smaller than g.
• Gravitational force follows an inverse square law

Surface Gravity Example

• Were you twice as far from Earth's center (8,000 miles from center instead of 4,000 miles), you would weigh one-fourth as much

Acceleration Due to Gravity on Different Planetary Bodies (g)

• Sun: 27.9
• Jupiter: 2.65
• Saturn: 1.05
• Earth: 1.0
• Venus 0.88
• Mars: 0.39
• Mercury: .38
• Moon: 0.16

Work

• Physics has a different meaning for work than in everyday usage.
• Doing work = applying a force to an object while moving it a given distance:
  • W = Fd
  • where F and d are magnitudes of the force and displacement.
• Work is a scalar (simple number), not a vector. There is no direction.

Work by a Constant Force

• Work (W) done on an object by a constant force (F) during a linear displacement along x-axis is W = F∆x.
  • F x is the x-component of the force and ∆x = xf – xj is the object's displacement.
  • Work is positive if Fx is positive and ∆x are both positive/both negative.
  • Work is done only by the force component || to the object's motion direction.

Work: Different Axis

• The work (W) done on an object by a constant force (F) during a linear displacement along the x-axis is W = Fd cos θ.
• Angle θ is the angle between forces F and ∆x.
• The components of the vector F:
  • Fx = F cos θ
  • Fy = F sin θ

Work: Various Axis

• Work can be determined when displacement isn’t along a specific axis.
• No work is done on moving an object horizontally if the applied force is perpendicular to displacement.
• Positive work is done when one lifts something from the floor (e.g. box). Negative work is lowering it.

Work Example

• 50 Kg: mass of a sled and salmon in total
• 1.20 × 10²N: magnitude of the force that the man exerts by pulling on the rope
  • How much work does he do on the sled horizontally on ground? @ 0=0 (and he pulls the sled)
    • W = Fdx = (1.20 × 102 N)(5.00 m) = 6.0 × 102J
  • How much work @ 0 = 30 and pulls the sled the same distance: -W = Fd cos 0 = (1.20 × 102 N) (5.00 m) cos 30 deg = 5.20 × 102 J
  • The coordinate position of 12.5m- and the man lets up applied force. A friction force of (45N) occurs btw ice and -sled (brings sled to rest @ position) 18.2 m - Q:much work does friction do on the sled Wfriction = Fx delta x = Fx ( Xf - Xi ) = ( −45.0 N (18.2 -12.5 ) N = =−2.6 × 102J

Friction, and the snow and kinetic friction

• Supposed that: in the previous example/ the Coefficient of Kinetic Friction (was in the btw loaded/ sled and snowy = 0.2) A. We want to find the work being done/ friction
• We want to find the network done (the man pulls a sled/ force is @ 1.20 × 102 N) @ zero deg.
• 0 we know that : Ʃ F = m ay and n-mg= 0 and then so =n = mg Fric =fk delta x= Un dx =0.2 5.0 Kg 9.8 N/kg (5.00m) fric = - 4.90*10^2 J Wnet = W app + W fric + Wnormal direction
• network done at 6.010^2 + and then add( - 4.9010^2=- Wnet=1.1*1^2

Work and Dissipative Forces

• Frictional work's very important every day to human-beings.
• The phrase/ work being done due to friction-the effect occurs on these processes (mechanical energy being alone)

Kinetic Energy and Work Energy Theorem

• Network (being done)/ on an object= how much change ( kinetic energy)
  • W net = KEf –KEi W met =f(net)* d cos(angular theta)=Mad
  • We are assuming that COS = 1
  • (Kinematics)- we are saying that : angular thetha = 1, with Fnd d (both being parallel)
  • V2= Vo+1 ad (kinematics equation)
    • We will say that – (½ M *squared - (½ mv2= )
    • The equation for kinematics (V2 = vo +2 Ad).

Conservative and Non Conservative forces:

• Conservative (IF) = Work. The number for whatever it does in moving = object being btw 2 points
• is also the same (no matter or whatever route you have actually taken, like with the path) =
• nonconservative= generally speaking/ dissapaitive=randomly disperses= bodies energy or even like friction/ propulsion forces) - W nc and W con = Triangle - (Potential And Energy).

Gravitational Potential Energy:

PEG= or to go against force ( gravitations) PEG work done against force) if the is/ lifting is happening (straight up)

• the needed/ force=is =the weight
• PEG= mgH
• angular Potential EN =Mgh

What are reference levels( for Gravitational Potential

(Potential) energy?

• solving problems/ potential you MUST choose( location= PE=energy).
• The choice is usually completely random- cause important is the (DIFFERENCE) in the potential And E.
• (Zero gravitational potential = is also/ the system can use reference levels )

Gravity & Energy Conservation

• Sum being constant =Energy - Always quantity that conserves. KE= I + PE=IF=kef _ pef
• Any system isolated w/only conservative forces - Will always == same