PHYSICS 121
*Week 1: Concepts of Motion & 1D Kinematics
MOTION DIAGRAMS (M-DIAGRAMS)
Physics is about Matter, Space, and Time → MOTION
Motion diagram ≡ pictorial representaon of moon
Particle Model: point represents an object
We can simplify by drawing arrows between points
Vector ≡ oriented line segment (𝑣, 𝑎, 𝑏
, etc.)
Adding vectors: add vert. & horiz. Components
Subtracting vectors: 𝑎 − 𝑏
= 𝑎 + (−𝑏
)
Negative vector is same length, opposite direction
Zero Vector ≡ no length or direction (𝑎 − 𝑎 = 0
)
Position vector: 𝑠; Displacement vector: ∆𝑠 = 𝑠− 𝑠
Time interval: ∆𝑡 = 𝑡− 𝑡; In m-diagrams, all ∆𝑡’s same
VELOCITY AND ACCELERATION
Speed ≡ how fast an object is moving
Speed A > Speed B if
A covers the same distance d in less time interval, or
A travels a longer distance d in the same time interval
Mathematically: 𝑎𝑣𝑔 𝑠𝑝𝑒𝑒𝑑 =
=
∆
On m-diagrams, the biggest ∆𝑟 has the biggest avg speed
Velocity ≡ vector w/ magnitude of speed (𝑣)
Average velocity: 𝑣 =∆
∆
Motion w/ constant 𝑣: length of vectors equal
Motion w/ changing 𝑣: length of vectors different
In rotational motion: 𝑣 is constantly changing
Objects interact with other objects move w/ changing 𝑣
Acceleration ≡ vector describing a change in velocity (𝑎)
Average acceleration: 𝑎 =∆
∆
Drawing Acceleration vectors on m-diagrams:
select two consecutive velocities
displace second vector to origin of first
draw acceleration from tip of first to tip of second
If 𝑣 is constant, 𝑎 = 0
If object speeds up, 𝑎&𝑣 are in the same direction
If object slows down, 𝑎&𝑣 are in opposite directions
If object moves along a curve, 𝑎 points “inside” the curve
COORDINATE SYSTEMS AND GRAPHS
Typical system: Positive x points right, positive y points up
Position and displacement measured in meters (m)
Time is measured in seconds (s)
Velocity measured in meters per second (m/s)
Acceleration measured in meters per second squared
(m/s2)
Graphs: abstract representation of motion.
x-axis is time, y-axis could be position, velocity, or
acceleration
SKETCHING PROBLEMS
Step 1 – Sketch the situation, showing objects as points at
beginning, middle (anywhere ∆𝑎), and end of motion
Step 2 – Draw a m-diagram
Step 3 – Choose and label a coordinate system
Step 4 – Place time, position, velocity at each point
Step 5 – Place accelerations between points
Step 6 – Make a table listing all known variables
Step 7 – Make a table listing all desired variables
Step 8 – Apply generic equations
INSTANTANEOUS VELOCITY
Instantaneous velocity is the slope of the 𝑠 vs 𝑡 curve
For uniform motion: rise over run (avg velocity)
For accelerated motion: derivative of curve
Mathematically: 𝑣(𝑡)= lim
∆→
∆
∆ =
()
INSTANTANEOUS ACCELERATION
Instantaneous acceleration is the slope of the 𝑣 vs 𝑡 curve
For uniform motion: acceleration is 0
For accelerated motion: derivative of curve
Mathematically: 𝑎(𝑡)= lim
∆→
∆
∆ =
()
=
()
*Week 2: 2D Kinematics
GENERIC EQUATIONS OF MOTION
(Only valid for constant acceleration)
𝑠= 𝑠+ 𝑣𝑡− 𝑡+1
2𝑎(𝑡− 𝑡)
𝑣= 𝑣+ 𝑎(𝑡−𝑡)
𝑣= 𝑣+ 2𝑎(𝑠− 𝑠)
FREE FALL
Use g = 9.8m/s2 for acceleration due to gravity
g points downwards (if positive y is up, 𝑎 = −𝑔)
Object goes down a plane angled θ from the horizontal
|𝑎|= 𝑔 × sin(𝜃)
VECTOR ARITHMETIC
Component Vector ≡ vectors parallel to x & y axes
𝚤 = 〈1,0〉; 𝚥 = 〈0,1〉
For a vector 𝐴
,
𝐴= |𝐴| × 𝑐𝑜𝑠(𝜃); 𝐴= |𝐴| × 𝑠𝑖𝑛(𝜃)
2D KINEMATICS & PROJECTILES
𝑠 = 𝑥𝚤 + 𝑦𝚥
𝑣 = 𝑣𝚤 + 𝑣𝚥
𝑎 = 𝑎𝚤 + 𝑎𝚥
For calculus, operate on each component individually
Relative velocity: 𝑣 = 𝑣 + 𝑣
For two moving objects, to find relative velocity, subtract
both velocities by the velocity of object one s.t. object
one’s velocity is 0
CIRCULAR MOTION
Angular velocity ≡ speed an angle changes (𝜔(𝑡) =
)
Unit: radians/s or degrees/s
Angular acceleration: (𝛼(𝑡) =
) unit: rad/s2 or deg/s2
Relationships between θ, ω, α identical to s, v, t
𝜃= 𝜃+𝜔 𝑡− 𝑡+1
2𝛼(𝑡− 𝑡)
𝜔= 𝜔+ 𝛼(𝑡−𝑡)
𝜔= 𝜔+ 2𝛼(𝜃− 𝜃)
Linear velocity ≡ the inst. 𝑣 of circular motion (tangent)
𝑣 = 𝜔𝑟 where r is the radius of circular motion
Period ≡ me it takes for a full rotaon (𝑇 =
)
For uniform circular motion, 𝛼 = 0
𝑎 ≠ 𝛼, since it always points inward & is changing
*Week 3: Forces and Motion
NEWTON’S THREE LAWS OF MOTION
Kinematics ≡ descripon of moon
Dynamics ≡ causes of moon
Force ≡ push or pull that acts on an object (𝐹
)
Agent ≡ object causing a force on another
Contact force acts by touching; long-range force does not
Force diagrams (F-diagrams):
Represent object as particle
Draw force vector arrow pointing in proper direction
Place tail of force vector on particle (not at tip)
Net Force ≡ vector sum of all forces on an object (𝐹
)
Mass ≡ property of objects; the resistance to 𝑎 (m)
Mass is measured in kilograms (kg)
Newton’s 1st Law: Objects at rest remain at rest; objects
moving remain moving at a constant 𝑣, iff 𝐹
= 0
Newton’s 2nd Law: 𝐹
= 𝑚 × 𝑎
Newton’s 3rd Law: Forces are interactions between
objects; each action-reaction pair is equal (magn.) and
opposite (dir.): 𝐹
= −𝐹
Force is measured in Newtons (N), = 𝑘𝑔 ×
FREE BODY DIAGRAMS (FBD)
Force types:
Name
Gravity
Spring
Tension
Normal
Agent
Earth
Springs
Ropes
Surface
Range
Long
Contact
Contact
Contact
Symbol
𝐹
𝐹
𝑇
𝑛
/
𝐹
Direct.
Down
Push/pull
Parallel
Perpend
Math
𝐹
=
𝑚
𝑔
|
𝐹
|=
𝑘
∆
𝑠
-
𝑚
𝑔
cos
θ
Name
Friction
Thrust
Drag
Agent
Sliding
Jet exhaust
Air/Fluid
Range
Contact
Contact
Contact
Symbol
𝑓
/
𝑓
𝐹
𝐷
Direct.
Opposite
Push/pull
Opposite
Math
𝐹
≤
𝜇
𝑛
𝑇
=
𝑣
𝐷
=
𝑣
𝜌𝐴𝐶
Force Identification:
Draw a circle around single object
Identify contact forces
Identify long-range forces
Draw a FBD
Drawing a FBD:
Choose a coordinate system
Represent object as particle at the origin
Draw identified forces, one by one
Compute 𝐹
by adding individual forces
Place 𝐹
vector next to diagram
Verify 𝐹
is consistent w/ 𝑎 in m-diagram
Correct FBD in case of discrepancies
GRAVITY, NORMAL FORCE, FRICTION
Force of gravity: 𝐹
= 𝑚𝑔
Normal force: Does not have a fixed magnitude; takes
whatever value is needed s.t. two touching surfaces have
the same acceleration:
Object sitting motionless on surface: 𝐹
= −𝐹
Object being pulled up by rope: 𝐹
= −𝐹
+ 𝑇
Tension overcomes, 𝑎 upward: 𝐹
= 0
Friction is either static (𝑓
) or kinetic (𝑓
)
𝑓
≤ 𝜇𝑛
; 𝑓
, = 𝜇𝑛
𝑓
= 𝜇𝑛
Friction opposes relative sliding of contact surfaces
*Week 4: Newton’s 3rd Law & Dynamics on a Plane
Action-Reaction Pair ≡ Two forces of N’s 3rd law (A-R pair)
Only one of the pair enters the free body diagram
For motion of an object, we only care about forces
exerted on it, not by it. N’s 3rd is important in systems of
interacting objects
Interaction diagrams:
Represent each object as a circle (incl. ropes &
pulleys); label each
Surface of earth & entire earth considered different
Draw a line for each interaction, label type of force
Identify the system
Draw FBD for each object in the system
Connect A-R forces w/ dashed line
Write N’s 2nd for each object. Solve simultaneous
equations, invoking N’s 3rd for A-R pairs
Conservation of mass: The 𝑚 of objects moving
together is ∑𝑚 (i.e., for 𝑚 and 𝑚,𝑚 = 𝑚+ 𝑚)
Composite object being pulled: F between is adhesion
𝐹
= 𝑚𝑎+ 𝑚𝑎, but 𝑎= 𝑎, so 𝑎 =
STRINGS, PULLEYS, CONSTRAINTS
*Week 5: Impulse, Momentum, and Energy
*Week 6: System Energy and Rigid Body Rotations
*Week 7: Universal Gravity and Oscillations
