Exam 2 Flashcards
1
Q
if a 25kg child climbs 2 steps of 20cm each, the work performance is?
A
W=mgh
25kg x 10 x .2
100Nm
2
Q
The form of energy associated with the stiffness and deformability of an object is called…
A
Strain energy
3
Q
The difference between positive and negative work stems from the direction of motion of the body the force acts on and the …
A
Direction of the force acting on the body
4
Q
When catching a baseball or softball with a glove…
A
The ball performs negative work on the glove, and the glove performs positive work on the ball
5
Q
A person performs a bench press. The bar’s was is 50kg. When this person’s arms are extended, the bar 0.6m above the starting point. How much work did the person do to push the bar off his chest to full extension of his arms?
A
W=mgh
50kg x 10 x .6
300Nm
6
Q
Kinetic energy is the energy…
A
due to motion
7
Q
Mechanical work is performed in all of these types of muscle activity (contraction) except for ?
A
isometric
8
Q
Power is equal to …
A
Force times displacement divided by time
Fxd / t
9
Q
Gravitational potential energy is the energy
A
Due to position relative to the earth
10
Q
What is the gravitational potential energy of a circus performer with a mass of 50kg walking on a tightrope 10 m above the ground?
A
U=mgh
50x10x 10
5000Nm
11
Q
The force to be considered in calculating the work you perform when climbing up a set of stairs is your…
A
Weight
12
Q
What is the net torque when you are holding a 10 lb weight 1 ft from your elbow (producing a clockwise torque) and your biceps is generating a 50 lb force with a moment arm of 3 in. (producing a counterclockwise torque)?
A
T=f x r
Tcw = 10 x 1=10
Tccw= 50 x 3in/12 (divide length by 12 to get ft) = 50 x .25 =12.5
Tnet = Tccw- Tcw
= 12.5-10
2.5ft-lb
13
Q
What happens to your center of gravity when you add a 200 N suitcase to your hand held out to your side?
A
The center of gravity shifts in the direction of the suitcase.
14
Q
An eccentric force…
A
does not pass through an axis of rotation
15
Q
Two forces acting on an object to cause only rotation of the object create…
A
A force couple
16
Q
The moment arm is defined as…
A
the perpendicular distance from the line of action of a force to a specified axis of rotation
17
Q
What is the torque if a 50 N force is applied 2 m from the axis of rotation?
A
T=Fxr
50 x 2
100Nm
18
Q
A force not directed through the center of an object (an eccentric force) tends to cause..
A
Both, linear acceleration & angular acceleration
19
Q
As a football player moves from a two-point stance (standing) to a four-point stance (hands and feet on the ground)…
A
the player’s stability is increased
20
Q
In the average female, the center of gravity is located…
A
lower than in the average male
21
Q
During angular motion, what is the relationship between the length of the moment arm and the torque created (when the force used is constant)?
A
The longer the moment arm, the greater the torque.
22
Q
An angle formed between two body segments is an example of…
A
A relative angle
23
Q
A gymnast dismounts the uneven parallel bars with some angular momentum about her transverse axis. Just after release, she is in a piked position. she then extends into a layout position before she lands. what happens as a result of this action?
A
Her moment of inertia decreases, and she rotates faster.
24
Q
An angle formed between a body segment and a horizontal reference line is an example of …
A
An absolute angle
25
Which plane is perpendicular to the longitudinal axis?
Transverse
26
What is the average angular velocity of an object moving through 30 degrees in 2 seconds?
15 degrees/second
27
Calculate the moment of inertia when an object's mass is 12kg and the mass is distributed 4 m from the axis of rotation.
12 x 4 squared =
192 kg x m squared
(I=mr squared)
28
A person riding a bicycle forward in a straight line would be moving in the ...
sagittal plane
29
For a rotating object, the acceleration directed toward the center of rotation is called
the centripetal acceleration
30
Movement of the hand at the wrist toward the ulnar side of the arm is ...
ulnar deviation
31
During a drive by golfer Ai Miyazato, the angular velocity of her club is zero at the top of the backswing and 30 rad/s at the bottom of the downswing just before impact with the ball. If the downswing lasts 0.30 seconds, what is the average angular acceleration of the club during the downswing?
a=changed w/ changed t
Final - initial: 30-0 / .30
30/.30
100 rad/s squared
32
Definition of work ...
transfer of energy by a force acting on an object as it is displaced
33
What is the SI unit of work?
Joule (J) = Newton-meter (N·m)
34
U=F(d)
Work formula
U = work done on an object
F = force applied to an object (N)
d = displacement of an object along line of action of the force (m)
35
To calculate work done on an object, what is needed:
you need to know the force applied, its direction, and how far the object moves in that direction
36
What is displacement?
Displacement is the change in position of an object, found by subtracting the starting position from the final position (final position - initial position)
37
Displacement Practice problem:
Weightlifter bench presses a 1000 N bar
* Bar starts 0.05 m above his chest
* Bar ends 0.75 m above his chest
* How much work on bar when raising and lowering?
* The average force: +1000 N
* Displacement: depends on ascent or descent
Ascent (raising bar)
* = +1000 N
* d = yf – yi = 0.75 m – 0.05 m = +0.70 m
* U = d = (+1000 N) × (+0.70 m) = +700 Nm = +700 J
Descent (lowering bar)
* = +1000 N
* d = yf – yi = 0.05 m – 0.75 m = –0.70 m
* U = d = (+1000 N) × (–0.70 m) = –700 Nm = –700 J
= +/-700 J
38
Definition of positive work ...
Positive work happens when a force moves an object in the SAME direction as the force, like when you throw a ball or jump up
39
Definition of negative work ...
Negative work happens when a force slows down or stops an object moving in the OPPOSITE direction, like when you catch a ball or land from a jump
40
Definition of positive MUSCLE work ...
Positive muscle work happens when a muscle SHORTENS as it contracts, pulling the attached bones in the same direction—like in a concentric contraction
EX: lifting a weight in a bicep curl
41
Definition of negative MUSCLE work ...
Negative muscle work happens when a muscle LENGTHENS while contracting, controlling movement in the opposite direction—like in an eccentric contraction
EX: lowering a weight in a bicep curl
42
Does an isometric contraction perform mechanical work?
Isometric contractions do not perform mechanical work because there is no movement, even though force is applied.
43
Energy definition ...
The ability to do work
44
Mechanical energy comes in what 2 forms:
1.) Kinetic energy
2.) Potential energy
45
Kinetic energy definition ...
Kinetic energy is the energy an object has because it is moving.
46
Kinetic energy formula ...
𝑲𝑬= 𝟏/𝟐 𝒎𝒗 squared
Where
* KE = kinetic energy (kg[m/s2]m) or Nm (Joule)
* m = mass (kg)
* v = velocity (m/s)
47
KE Practice problem:
How much KE in a 5-ounce baseball (145 g) thrown at 80 miles/hour (35.8 m/s)?
* mass = 145 g (0.145 kg)
* velocity = 35.8 m/s
𝑲𝑬= 𝟏/𝟐 𝒎𝒗 squared
KE= 1/2 (0.145kg)(35.8) squared = 92.9 Joules
48
Potential energy definition ...
Potential energy is stored energy that an object has because of its position or shape
49
What are the 2 forms of potential energy (PE) ?
1.) Gravitational Potential energy
2.) Strain Potential energy
50
Gravitational Potential energy definition ...
Gravitational potential energy is the stored energy an object has because of its height above the ground
EX: Rock sitting on top of a hill. The higher the rock, the more potential energy it has because of its position relative to the ground
51
Strain Potential energy definition ...
Strain potential energy is the energy stored in an object when it is stretched, compressed, or deformed, like a compressed spring
EX: A stretched rubber band. When you pull on it, the energy is stored, and it can be released when you let go.
52
Gravitational PE formula ..
PE=mgh (mass × gravity × height)
Where
* PE = gravitational potential energy of an object (Nm = J)
* mg = mass or weight (N)
* h = elevation (height) above reference point (m) (ground or some
other surface)
53
Gravitational Practice problem:
How much PE in a 100 kg held 70 cm above the starting
point on the weightlifter’s chest?
Use 5 cm above lifter’s chest as the reference point
m = 100 kg
g = 10 m/s2
h = 70 cm = 0.70 m
PE = mgh = (100 kg) (10 m/s2) (0.70m) = 700 J
54
Strain energy formula ...
SE = ½k△x squared
Where
* SE = strain energy stored in a deformed object (Nm or J)
* k = stiffness constant of the material (N/m)
* △x = deformation (change in length) of the object (m)
55
Work–Energy Principle definition ...
The net work done by all the external forces acting on an
object (or system) causes a change in the mechanical
energy of the object which includes both kinetic energy (motion) and potential energy (position)
In simple terms: The work done on an object changes its energy, either by increasing its motion or altering its position.
56
Work–Energy Principle formulas ...
The formula expresses how the total change in mechanical energy (d) is the sum of the changes in kinetic energy (△KE) and potential energy (△PE):
d = total mechanical energy△
d = △KE + △PE
d = (KEf – KEi) + (PEf – PEi)
57
Conservation of Mechanical Energy definition ...
Conservation of Mechanical Energy means that if ONLY gravity is acting on an object (no other external forces), its total mechanical energy (kinetic + potential energy) stays the same
58
Conservation of Mechanical Energy formula ...
(KEi + PEi + SEi) = (KEf + PEf + SEf)
* Where
* KE = kinetic energy
* PE = potential energy
* SE = strain energy
* i,f = initial and final
59
Conservation of Mechanical Energy examples
As a moving object rises up in the air
* Kinetic energy decreases (velocity decreases)
* Potential energy increases (height increases)
* Total energy remains constant
as a falling object gets closer to the ground
* Kinetic energy increases (velocity increases)
* Potential energy decreases (height decreases)
* Total energy remains constant
60
Power definition ...
Power is the speed at which work is done, or energy is transferred (How quickly or slowly work is done)
61
Power formula ...
P= U/Δt
Where
* P= average power
* U = work (d)
* △t = time taken to do the work
62
Power units ...
J/s = Watt
One watt equals one joule per second (J/s).
63
Power: Alternative Calculation
Power can also be calculated by multiplying force and velocity:
𝑃=𝐹×𝑣
Where:
P is power (in watts, W)
F is force (in newtons, N)
v is velocity (in meters per second, m/s)
64
Power output ...
Power output can be the same even if the force and velocity are different, as long as their product (force × velocity) is the same
In simple terms: You can get the same amount of power whether you use a lot of force with slow speed, or little force with fast speed
65
Velocity of locomotion formula
Velocity of locomotion = cycle rate × cycle length
If a runner takes 80 steps per minute (cycle rate) and each step moves them 1.2 meters forward (cycle length), their speed is:
80 x 1.2 = 96 meters per minute
66
Cycle rate definition
How often the motion is repeated per minute
67
Cycle length definition
What distance is traveled per cycle
68
Velocity of locomotion is...
determined in activities that involve repeated movements, like walking, running, cycling, or swimming.
69
power & locomotion
Power is the ability to generate force quickly, which helps with speed and strength in movement.
Locomotion is the ability to move from one place to another, like walking, running, cycling, or swimming.
Together, power and locomotion describe how force and movement combine determine how fast and efficiently a person or animal moves.
70
To move faster or slower, performer can change ...
Cycle rate, cycle length, or both or have them changed by injury or disease
71
Power Constraints on Human
Performance ...
Power output and endurance are inversely related:
* High power output sustainable for shorter duration (If you move with high intensity - sprinting, you can’t keep it up for long)
* Low power output sustainable for longer duration (f you move with low intensity- jogging, you can keep going much longer)
72
Torque definition
Torque is the twisting force that makes something rotate.
It is also called moment of force or just moment.
Torque makes things spin or turn, like a door swinging open or a bike pedal moving.
Your muscles create torque to move your joints—like your arm rotating when you throw a ball.
73
What are the 3 different types of force?
-Centric
-Eccentric
-Force couple
74
Centric force definition
When a force is applied directly through the center of an object (center of gravity), it only makes the object move in a straight line (translation). For example, pushing a box from the center will make it slide forward.
75
Eccentric force definition
When the force is applied off-center (not through the center of gravity), it makes the object move in both a straight line and rotate. For example, pushing a door near the edge will make it swing open (rotate) and move in the direction of the push.
76
Force couple definition
2 eccentric noncolinear forces, equal and opposite in direction (Causes changes in rotation only)
77
What causes changes in translation only?
Centric force
78
What causes changes in translation and rotation?
Eccentric force
79
What causes changes in rotation only?
Force couple
80
moment arm definition
A moment arm is the shortest distance from the line of force to the point where the object rotates, helping to determine how much the force will cause rotation.
81
Represented with r ?
Moment arm
82
moment arm (longer, shorter)
The moment arm measures how far a force is from the rotation point, with a longer moment arm making it easier to turn something, and a shorter one making it harder.
83
Torque formula
T = F × r
Where
* T = torque
* F = force (magnitude and direction)(N)
* r = moment arm (m)
84
SI units of torque
Nm
85
What do you need to fully describe torque?
You need to know the axis of rotation, the strength of the force, the direction it turns (clockwise or counterclockwise), and how it relates to the body’s movement.
86
Applications of Torque
Torque can be created in different ways: using a large force with a short moment arm or a small force with a long moment arm. T
Tools help humans apply more torque by increasing the distance (moment arm) where the force is applied, making it easier to turn or lift objects.
87
Muscular Torque
Muscle produces a pulling
force (Fm), crosses a joint, and has a moment arm about the joint axis
Muscle force creates
torque at a joint helping to move it.
88
How does changing the Joint Angle
Changes the Muscle Moment Arm?
As a joint moves, the length of the muscle's moment arm changes, affecting how much force it can produce.
89
What does fm represents?
Muscle force
90
Strength Training Torques
In strength training, weights create resistance that changes as you move through range of motion. resistance is stronger (increase) when distance from the joint (moment arm) is longer and weaker (decrease) when it's shorter. Your muscle torque works against this resistance to lift the weight.
91
Strength Training Devices and Torque
In strength training machines, the resistance force comes from weights, and the resistance torque changes as the machine moves through ROM. This happens because the distance (Perpendicular) from the machine's axis to the cable (resistance moment arm) changes throughout the movement.
92
Static equilibrium definition
Static equilibrium means that all forces and all torques acting on an object must balance out to zero, so the object stays still without moving.
ΣF = 0
* Sum of forces = 0 in all directions
* ΣT = 0
* Sum of torques = 0 around all axes
93
Net torque definition
The net torque is the sum of the individual torques.
Torques that act around the same axis can combine by adding if they rotate in the same direction and subtracting if they rotate in opposite directions.
94
Net torque formula
Net torque is found by adding all torques acting on an object, counting clockwise torques as negative and counterclockwise torques as positive.
ΣT = Tclockwise + Tcounterclockwise
95
Net torque example problem
Imagine a seesaw with a 10 N force pushing down on one side 2 meters from the pivot and a 15 N force pushing down on the other side 1 meter from the pivot.
Torque from the 10 N force: 10×2=20 N·m (clockwise)
Torque from the 15 N force: 15×1=15 N·m (counterclockwise)
Net torque: Σ𝑇=𝑇counterclockwise−𝑇clockwise=15−20=−5Nm
Since the net torque is negative, the seesaw rotates clockwise.
96
Net torque clockwise =
NEGATIVE
97
Net torque counterclockwise =
POSITIVE
98
rotational equilibrium, where the net torque is zero meaning...
if the torques balancing each other are equal in size but opposite in direction, the object stays still and does not rotate.
99
Center of gravity definition
The center of gravity (CG) is the point where an object's weight is evenly balanced.
cg is an imaginary point in space
* Not a physical entity
* Not a fixed point (The CG can move if the shape or position of the object changes. For example, if you shift a weight on a board, the CG of the board will move too.)
* The force of gravity acts downward through the cg
100
Center of gravity example
Think of it like a seesaw: if you put the right amount of weight on each side of the seesaw at the right distance from the middle, the seesaw will be balanced. The middle of the seesaw is like the CG—it's where all the weight is balanced evenly.
101
Locating the Center of Gravity of an Object
Locating the center of gravity (CG) of the human body means finding the point where all the body parts balance each other. It's like the balance point of the body. The CG is where the total sum of the torques (rotational forces) from all the body parts equals zero, meaning the body is perfectly balanced around that point.
102
Mathematically find the center of gravity (CG) of an object
ΣT = Σ(W × r) = (ΣW) × rcg
W: The weight of each part of the object.
r: The distance from a reference point (like a pivot or axis) to where the weight acts.
ΣW: The total weight of the entire object.
rcg: The distance from the reference point to the center of gravity (CG), which we're trying to find (moment arm)
103
The center of gravity (CG) of the human body changes depending on how you move your limbs:
CG location depends on limb position: Where your body’s balance point is will change based on how your arms, legs, and other body parts are positioned.
CG height: For women, the CG is about 55% of their height, and for men, it’s around 57%.
Lift your left arm away from your side: If you raise your left arm, your body’s balance point (CG) moves slightly to the left.
Lift your left arm in front: If you raise your left arm in front of you, your balance point shifts forward.
Raise both arms away from your side: If you raise both arms, your CG shifts upward.
So, the CG moves depending on how your body parts are positioned, and these changes can make you feel off balance.
104
COG example
Imagine a 2-meter-long beam with two weights:
A 10 N weight is 0.5 meters from the pivot.
A 20 N weight is 1.5 meters from the pivot.
To find the center of gravity (CG):
Calculate the torque of each weight:
Torque from 10 N weight: 10×0.5=5Nm
Torque from 20 N weight: 20×1.5=30Nm
Add the torques together:
ΣT=5+30=35Nm
Add the weights together to get the total weight:
ΣW=10+20=30N
Use the formula ΣT=(ΣW)×r cg to find the location of the CG:
35=30×r cg
𝑟 𝑐𝑔=35/30=1.17meters
So, the center of gravity is located 1.17 meters from the pivot point.
105
Center of Gravity and Performance
When you jump or throw something, your body behaves like a projectile, meaning the center of gravity (CG) follows a predictable curved path. While the CG moves in that path, the parts of your body (like your arms and legs) can move in different directions to help with balance or control. For example:
Jump and reach: You can reach higher with one hand than with two because the CG is more focused, and you have more control with one arm, allowing your body to adjust better.
Hang time: When you're in the air, your arms and legs may move in opposite directions to help you stay balanced or look like you're suspended in the air for longer.
106
Center of Gravity and Stability
Stability refers to how easily an object can be tipped over or thrown off balance. An object is more stable if it can return to balance when disturbed and is harder to move. The stability of an object depends on:
Height of center of gravity
Size of the base of support
Weight of the object
an object that is low to the ground, wide, and heavy will be more stable and less likely to fall over.
107
Angular Kinematics definition
Angular kinematics is the study of how things rotate. It looks at how objects move in circles or along curved paths around a point (called the axis).
108
Descriptors of angular motion
How far?: This describes how much an object has rotated or how far it has turned, usually measured in degrees or radians.
What direction?: This tells you which way the object is rotating—clockwise or counterclockwise.
How fast?: This refers to how quickly the object is rotating, which can be measured in terms like angular velocity (how many degrees or radians it turns per second).
Speeding up, slowing down?: This describes whether the rotation is getting faster (acceleration) or slower (deceleration).
So, angular kinematics focuses on understanding the motion of things that rotate—how far they rotate, which way they go, how fast they rotate, and if their speed is changing.
109
What is an angle ?
Intersection of two lines, two planes, or a line and a plane
* Angle measures the orientation of the lines and/or planes
* An angle is represented by the Greek letter ⍬ (theta)
110
Units of an angle...
Degrees (°): A full circle is 360°
Radian (rad): 1 radian is about 57.3°
Revolution: A full rotation or twist, which is 360° rotation
111
Angular position definition
the orientation of an object relative to a reference position
112
Angular position types (2)?
Absolute angular position & Relative angular position
113
Absolute angular position
This is when one line or plane stays fixed (like the earth), and we measure the angle of another object or body part in relation to it.
ex: Imagine you’re standing straight up, and your arm is at a certain angle compared to the ground. The ground is the fixed point, and your arm is the moving part. The angle between your arm and the ground is your absolute angular position. The ground doesn’t move; only your arm does.
114
Relative angular position
This is when both lines or surfaces can move.
imagine moving your arm and your lower arm. The angle between your upper arm and lower arm at the elbow is the relative angular position. Both parts of the body (upper and lower arm) can move, so this is a relative position because the angle depends on how the two parts are positioned in relation to each other.
115
Angular displacement definition
Angular displacement is the change in absolute angular position experienced by a rotating line
Angle between the final and initial angular positions: Δθ=θf−θi
116
Angular displacement formula
Δθ=θf−θi
Direction matters:
-Clockwise rotation is negative (-)
-Counterclockwise rotation is positive (+)
117
Angular displacement example
Imagine a spinning wheel:
The wheel starts at 30° (θᵢ = 30°).
It rotates counterclockwise to 90° (θ𝒻 = 90°).
The angular displacement is:
Δθ=θf−θi=90∘−30∘=60∘
Since the rotation is counterclockwise, the displacement is +60°
118
Right -Hand rule
The Right-Hand Rule helps determine the direction of rotation:
1.) Find the axis of rotation (real or imaginary line around which something spins).
2.) Identify the plane of motion (the surface in which the object moves).
3.) The axis is always perpendicular to the plane of motion (at a 90° angle).
4.) Use your right hand:
Point your thumb in the positive direction of the axis. Your fingers curl in the positive direction of rotation.
119
Angular and Linear Displacement definition
When something rotates, points farther from the axis move more than points closer to it.
Ex: A baseball bat makes the ball travel farther because the end of the bat moves more than the hands holding it.
120
Angular and Linear mathematically
ℓ = △⍬r
* Where
* ℓ = arc length (distance travelled, in meters)
* △⍬ = angular displacement (must be in radians)
* r = radius (distance from axis of rotation, in meters)
Imagine a wheel with a radius of 2 meters:
It rotates 1 radian (about 57.3°).
The distance traveled along the edge is: ℓ=1×2=2 meters
The farther from the center (larger r), the more distance is covered for the same rotation.
121
angular velocity definition
how fast something is rotating (Rate of change of angular position)
122
angular velocity mathematically
ω= Δθ/Δt
Where
* = angular velocity (Note: in rad/s if △⍬ measured in radians)
* △⍬ = angular displacement
* △t = time
A fan blade spins 90° in 2 seconds.
Convert 90° to radians (π/2 rad).
Calculate:
ω=π/2 /2=π/4 rad/s
Key Idea:
Faster rotation = higher angular velocity.
123
Instantaneous Angular Velocity
Rate of rotation at an instant in time
Think of a speedometer in a car:
It shows the car’s speed right now, not over time. Instantaneous angular velocity does the same for rotation.
124
Average angular v vs instantaneous angular v
Average angular velocity = speed over a period.
Instantaneous angular velocity = speed at an exact moment.
125
Angular Velocity and Linear Velocity
Points farther from an axis of rotation travel greater linear
path than points closer to an axis of rotation at the same time
* Farther: greater distance (and greater displacement)
Points farther from an axis of rotation travel at a greater
linear speed (and greater instantaneous linear velocity)
than points closer to an axis of rotation
126
Angular Velocity and Linear Velocity mathematically
vT = r⍵
Where
* vT = Instantaneous linear velocity tangent to circular path (m/s)
* ⍵ = instantaneous angular velocity (must be in rad/s)
* r = radius of rotation
A wheel spins at 5 rad/s, and a point is 0.4 m from the center.
v=(5 rad/s)×(0.4 m)
v=2 m/s
A point 0.4 m away moves at 2 m/s in a straight-line path.
If the point was farther (0.8 m), it would move faster:
v=(5×0.8)=4 m/s
Farther from the axis = Greater speed!
127
Tangential Linear Velocity definition
how fast and in what direction a rotating point moves at an instant.
It is always tangent to the circular path (perpendicular to the radius).
128
Tangential Linear Velocity formula
v=r⋅ω
Where:
v = Tangential linear velocity (m/s)
r = Radius (m) (distance from the axis)
ω = Angular velocity (rad/s)
A bike wheel spins at 6 rad/s, and a point is 0.5 m from the center.
Find tangential velocity:
v=(0.5)×(6)=3 m/s
The point moves at 3 m/s in a straight line tangent to the wheel.
129
Angular Acceleration definition
How fast angular velocity is changing over time. Happens when something starts, stops, speeds up, slows down, or changes direction.
130
Angular Acceleration direction
Speeding up or starting
* Angular acceleration is in the direction of angular motion
Slowing down or stopping
* Angular acceleration is in the direction opposite of angular motion
Sign of velocity and sign of acceleration the same
* Body is rotating faster (speeding up)
Sign of velocity and sign of acceleration the opposite
* Body is rotating slower (slowing down)
131
Angular Acceleration formula
α= Δt/Δω
A spinning top speeds up from 2 rad/s to 6 rad/s in 2 seconds.
Find angular acceleration:
α=6−2 / 2 = 4/2=2 rad/s²
The top has an angular acceleration of 2 rad/s².
132
Types of Angular and Linear Accelerations
Angular Acceleration (α)
Tangential Linear Acceleration (aₜ)
Centripetal (Radial) Acceleration (a_c)
133
Angular and Linear Acceleration
When angular velocity increases, the Tangential velocity of points on the body increase
134
Tangential acceleration
component of linear acceleration that is along the circular path or tangent to the circle. It represents how quickly a point’s speed along the circular path changes.
types of Tangential Acceleration:
Positive Tangential Acceleration: When the object is speeding up along its circular path.
Negative Tangential Acceleration: When the object is slowing down along its circular path.
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Tangential acceleration formula
at=r⋅a
Where:
aₜ = Tangential acceleration (m/s²)
r = Radius of the circle (m)
α = Angular acceleration (rad/s²)
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Centripetal (radial) acceleration
acceleration that points towards the center (radius) of the circular path. It keeps the object moving in a circle by constantly changing its direction.
Caused by the centripetal force (towards the center of the path)
always directed toward the center of the rotation. It doesn't affect the speed but changes the direction of the object’s motion as it moves in a circle.
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Centripetal (radial) acceleration continued
Linear acceleration: occurs when an object slows down,
speeds up, or changes direction
The Tangential velocity of an object moving on a circular path at a constant Angular velocity changes direction from point to point
* If released at different points on the path, object will travel linearly
in different directions
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a T=rα
This formula represents the relationship between tangential acceleration (aₜ), the radius (r), and angular acceleration (α).
Centripetal (Radial) Acceleration (aᵣ)
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Two Equations for Centripetal (Radial) Acceleration (aᵣ)
Centripetal Acceleration in Terms of Tangential Velocity (vₜ):
ar=v2T/r
Where:
aᵣ = Centripetal (radial) acceleration (m/s²)
vₜ = Tangential velocity (m/s) — how fast the point moves along the circular path
r = Radius of the circular path (m)
This equation tells us that the centripetal acceleration depends on the square of the tangential velocity and is inversely proportional to the radius.
2. Centripetal Acceleration in Terms of Angular Velocity (ω):
ar=r⋅ω2
Where:
aᵣ = Centripetal (radial) acceleration (m/s²)
r = Radius (m)
ω = Angular velocity (rad/s) — how fast the object is rotating
This equation shows that the centripetal acceleration is directly proportional to the radius and the square of the angular velocity.
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Centripetal Acceleration
If ⍵ is constant, ar is directly proportional to the radius
If vT is constant, ar is inversely proportional to the radius
ar greater running on inside lane compared to outside lane
There must be a force causing the change in linear velocity
* Object on a circular path: centripetal force (toward the center)
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Anatomical Planes
sagittal plane : right and left halves
frontal plane: anterior and posterior halves
Transverse plane: superior and inferior halves
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sagittal plane movements
flexion
extension
dorsiflexion
planter flexion
(running, walking, or bending at the waist)
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Frontal plane movements
abduction
adduction
lateral flexion
elevation & depression
inversion & eversion
(Jumping jacks, lateral shoulder raises, side lunges)
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Transverse plane movements
rotation
internal (medial) rotation
external (lateral) rotation
pronation
supination
(Swinging a golf club, throwing a baseball, twisting sit-ups)
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Anatomical Axes
Anteroposterior (AP) axis: (cartwheeling)
* Imaginary line running from anterior to posterior (sagittal axis) Perpendicular to a frontal plane
Transverse axis: (somersaulting)
* Imaginary line running from left to right (frontal axis) Perpendicular to a sagittal plane
Longitudinal axis: (twisting)
* Imaginary line running superior to inferior (vertical axis) Perpendicular to a transverse axis
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planes motion and axes
frontal/transverse axis correlates with sagittal plane
sagittal/anterposterior axis correlates with frontal plane
vertical/longitudinal axis correlates with transverse plane
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Identifying Planes and Axes of Motion
A body moves in a specific direction within a plane, and rotates around an axis that is always at a right angle (perpendicular) to that plane.
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joint actions
Joint actions describe how two connected body parts move relative to each other, usually in terms of angles, within a plane and around an axis.
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Radial deviation
hand moves toward the thumb side
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ulnar deviation
hand moves toward the pinky side
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Circumduction
circular movement of body part, where the segment moves around multiple axes, typically involving the transverse and anterior-posterior (AP) axes. This motion creates a cone-shaped path, like when you move your arm in a circular motion at the shoulder joint.
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angular kinematics
Study of motion without considering the forces that cause the motion. Describes how an object moves in terms of angular displacement, angular velocity, and angular acceleration.
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Angular Kinetics
study of forces and torques that cause angular motion. understanding the cause of angular motion, including the relationship between torque (force that causes rotation) and angular acceleration.
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Inertia definition
resistance of a body to changing motion
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Angular inertia definition
property of an object to resist changes in its angular motion
Unlike linear inertia, which depends only on mass, angular inertia (moment of inertia) depends on both mass and how that mass is distributed relative to the axis of rotation.
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Moment of inertia
measures how difficult it is to rotate an object.
If the mass is farther from the axis, rotation is harder (more resistance).
If the mass is closer to the axis, rotation is easier (less resistance).
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Moment of inertia formula
Moment of inertia (Ia ) is the total resistance to rotation, considering all mass particles and their distances from the axis.
Formula: Ia=∑mir2
What it means:
Each mass (mi) contributes to rotation resistance.
The farther a mass is from the axis (ri), the greater its effect.
Closer mass = less resistance, farther mass = more resistance.
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radius of gyration
represent how mass is distributed in a rotating object. (makes it easier to calculate I)
It tells us how far from the axis all the mass would need to be concentrated to produce the same rotational resistance (moment of inertia) as the object’s actual shape.
Instead of calculating every mass particle’s distance, we use this single value to estimate the object’s resistance to rotation.
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radius of gyration Mathematically
Ia = mka2
Where
* Ia = moment of inertia about axis a through the cg
* m = mass of the object (kg)
* ka = radius of gyration about axis a through the cg (m)
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moment of inertia (I)
depends on both mass and how the mass is spread out from the axis of rotation.
Mass effect: If the mass doubles, the moment of inertia doubles (2× I).
Distribution effect: If the mass is spread twice as far from the axis (radius of gyration doubles), the moment of inertia quadruples (4× I).
Key idea: Mass farther from the axis increases rotational resistance much more than just adding more mass.
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Moment of Inertia About Eccentric
Axes
Not all rotation occurs around the cg
* Eccentric axis: Implement axis not passing through cg
Performer holds “grip” or “end” of implement, not cg (Bat, hammer, stick, racket)
New axis is called a parallel axis
* Mass farther from axis of rotation means I is greater
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Moment of Inertia about Eccentric Axes Mathematically
Ib = Icg + mr2
Where
* Ib = moment of inertia about axis b
* Icg = moment of inertia about axis through the object’s center of
gravity and parallel to axis b
* m = mass of object
* r = radius = distance from axis b to parallel axis through the
center of gravity
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Moments of Inertia in the Human
Body
Three principal axes in the human body
* Anteroposterior (cartwheel)
* Transverse (somersault)
* Longitudinal (twist)
The moment of inertia around each axis depends on the position or orientation of the limbs relative to the axis
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Manipulating the Moments of Inertia
of the Human Body
You can change the moment of inertia (I) by adjusting body position or limb angles because this changes mass distribution around a joint.
Changing the angle of multiple joints shifts the mass distribution around the center of gravity (cg), affecting the moment of inertia (Icg). (Figure skaters bring their arms close to their body while spinning to reduce Icg, allowing them to spin faster.)
Example 1: A sprinter bends their leg during the recovery phase to bring mass closer to the hip, reducing I and making leg movement faster.
Example 2: When swinging a tool, gripping it closer to the head reduces I, making it easier to swing, while holding it at the end increases I, requiring more effort to rotate.
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Moments of Inertia and Linear
Velocity
A longer radius means a greater tangential velocity (vT) and a greater moment of inertia (I).
When choosing the length of something to swing (like a bat or racket), there's a trade-off:
A longer implement is harder to swing (greater I), but it can generate more velocity.
A shorter implement is easier to swing (lower I), but the velocity might be lower.
Equipment design (length, mass, and how mass is distributed) affects both ease of swing and effectiveness.
Technique changes (like choking up on a bat) reduce the radius, making it easier to swing while sacrificing some velocity.
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Angular momentum
Angular momentum measures an object's rotational motion, describing both how it is currently moving and how difficult it would be to change its motion.
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Angular momentum formula
Ha = Iaωa
Where
* Ha = Angular momentum about axis a (kg∙m2/s)
* Ia = moment of inertia about axis a (kg∙m2/s)
* ωa = angular velocity about axis a (must be in rad/s
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Vector quantity
Vector quantity: magnitude and direction
* Magnitude: Iω (units: kg∙m2/s)
* Direction: same as direction of ω
* Follows right-hand rule conventions
Ha will change because of changes in either or both of Inertia and angular momentum. Unlike linear momentum where velocity varies, mass constant in human movement
both I and ω can change, affecting angular momentum.
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Angular Momentum of the Human
Body
When multiple limbs rotate at different angular velocities (ω), the total angular momentum (Ha) of the body is the sum of the angular momenta of each individual segment of the body.
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total angular momentum (Ha) formula
Ha ≈ (Ʃ Ii/cg ωi)
Where
* Ha = angular momentum about axis a through the cg
* Ii/cg = moment of inertia of segment i about the entire body cg
* ωi = angular velocity of segment i
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Newton’s First Law: Angular Interpretation
Newton’s First Law: Angular Interpretation states that the angular momentum of an object stays the same unless a net external torque acts on it.
If the moments of inertia (I) of the object remain constant, then the angular velocity (ω) will also remain constant.
In simple terms, without external forces trying to change the rotation, an object will keep rotating at the same speed and in the same way.
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rigid projectile
For a rigid projectile, gravity acts through the center of gravity (cg), and since the weight has no distance from the cg, it creates no external torque, meaning there is no external force affecting the body's rotation.
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conservation of angular momentum
H (product of I and ω) remains constant unless a net external torque acts on the rotating body
Mathematically: Hi = Iiωi = Ifωf = Hf
Where
* Hi = initial angular momentum
* Hf = final angular momentum
* Ii = initial moment of inertia
* If = final moment of inertia
* ωi = initial angular velocity
* ωf = final angular velocity
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Human body is a system of rigid links with modifiable I
With H constant (no external torque applied)
* Increase in I, there must be a proportional decrease in ω
* Decrease in I, there must be a proportional increase in ω
* For faster spin, reduce I (tuck, arms in)
* For slower spin, increase I (layout, arms out)
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Control angular velocity of Limbs or Trunk
Hcg remains constant while a projectile
* Individual segments have Hi/cg from I and ω
* Repositioning one or more segments of body must be
countered by repositioning one or more segments in the
opposite direction to maintain constant Hcg
* Repositioning means changing ω
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Newton’s Second Law: Angular
Interpretation
If a net external torque acts on a body, the body will angularly accelerate in the direction of the net external torque; its acceleration will be directly proportional to the
torque and inversely proportional to its moment of inertia
* Similar to Newton’s second law for linear motion
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Newton’s Second Law: Angular
Interpretation formula
ƩTa = Iaαa
Where
* ƩTa = net external torque about axis a
* Ia = moment of inertia of the object about axis a
* αa = angular acceleration of the object about axis a
Remember that both I and α can change or be changed!
(continued)
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Angularly accelerate
change in angular momentum
The change in angular momentum of an object is proportional to the net external torque exerted on it, and
the change is in the direction of the net external torque
* Torquea and ωa have same sign: Ha increases
* Torquea and ωa have opposite sign: Ha decreases
* If I changes when T applied, change in ω is greater
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Impulse–Momentum: Angular Version Mathematically
Where
* Ʃa = average net external torque about axis a
* ΔHa = change in angular momentum about axis a
* Hf = final angular momentum about axis a
* Hi = initial angular momentum about axis a
* Δt = change in time
* If I changes during , the change in angular velocity (ω) is greater
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Newton’s Third Law: Angular
Interpretation
For every torque exerted by one object on another, the other object exerts an equal torque back on the first object but in the opposite direction (Equal but opposite)
* Torque on each body is of equal magnitude, not the effect
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