AP PHYSICS UNIT 7 ROTATIONAL MOTION & ANGULAR MOMENTUM (Torque)
What's the mistake here?
I" should be on the bottom because I on the top means bigger I leads to a bigger W but that is false.
The angular momentum of an object in a circle can be calculated using
I* W or M*V*R
Turning effect of a doorknob is dependent on
Radical distance Farther away from the hinges (greater torque) we push the easier it is to move.
Which of the following statements about T1 and T2 is correct? (Atwood scenario)
T2 > T1 because an unbalanced clockwise torque is needed to accelerate the wheel clockwise.
Torque equation
T=Fr r= full length of lever or radical distance F= force perpendicular
"I" rotational inertia is proportional to
Radias squared I= m * r^2
What influence moment of inertia
Radias, Mass
Practice problem 1 what went well?
- Using the right equation to find time - doing good conversions - conceptual physics to see what's positive and what is negative
If there is no net force then what does change in P equal
0
If there is no net torque then what is the change in angular momentum?
0
How do we find out how many degrees 0.59 revolutions is?
0.59 x 360 = 212°
In a spinning wheel consisting of 36 numbers what degree is each slice?
10° to get 360 total
How many radians are in one revolution
2π
How many degrees is one revolution
360°
How far did the wheel rotate?
900mm From red spoke to full rotation
How many newtons of force to keep balance and why?
90N because the left side multiplies to 360 so therefore the right must be 360 (90x4) And think if distance goes up on one side then force must go down to balance. Since it's a multiplication scenario for torque.
A net torque applied to a system over time will result in
A change in angular momentum
MC practice 2 slopes
A is correct because the slope is constant enough Luke the student claims.
How do we answer part A of this FRQ?
A larger force will result in the top spinning faster. The string will unwind faster which will decrease the time. Remember a stronger torque/ force will result in a greater acceleration which means less time.
What causes centripetal acceleration?
A net force must be exerted by external agents toward the circle's center to cause centripetal acceleration. Ex: friction on a spinning disk for a ladybug
Rotational inertia
A resistance to rotating
How do we answer part B of this FRQ? Focused on moment of inertia
A smaller rotational inertia and the same force (torque) will result in the top spinning faster. The string will unwind faster which will decrease the time
Friction can count as
A torque
centripetal acceleration angular
Ac = w^2 * r Angular speed squared times radias
Trust yourself in physics I got this graph correction correct for E! For which part of the graph either the time before T or after T (when friction is present) is incorrect
After T because friction would cause it to slow down so the acceleration graph should be a parabala slowing down not a linear line upward.
Argument for a collision object that bounces off...
After the bouncy collision, the disk has angular momentum in the clockwise direction. To keep the system angular momentum constant the magnitude of the rods counterclockwise angular momentum must be greater than before.
Analyzing experimental design data
Alpha vs torque graph We manipulated the torque and found a linear relationship. Best fit line
How do we derive an equation for before the collision with the rod and post collision?
Always start with Li= Lf
A torque causes a rotation which causes
An angular acceleration
What happens when there is friction from torque
An external torque of friction will act in the opposite direction of the moving top and as a result, the angular momentum will decrease.
axis of rotation
An imaginary line around which rotation occurs In the center
Objects translating about some external pivot or reference point have
Angular momentum L= m*v*r r= perpendicular distance I = mr^2 w= v/r Ex: airplane moving around me
As a weightlifter in a chair pulls dumbbells closer to them what happens to the angular momentum of the chair and person?
Angular momentum will stay the same because there is no external torque.
When the student jumps on a merry go round their variables become
Angular/ rotational use R^2
How do we maximize a torque?
Apply the force perpendicular to the radical axis at the farthest distance from axis of rotation.
Torque vs time graph
Area= change in angular momentum
In experimental design for the ap test we should be
As clear as possible
Experimental design example
Attach a string to the edge of the object and use the force sensor to measure how much force is applied to the string. Multiply the Result by the radias to determine the torque. Record Once the object is rotating we will use the rotational motion sensor to get the angular acceleration Repeat each step varying the amount of force applied in each trial (multiple trials) We will keep the force as consistent as possible while pulling to reduce error
Why is option A and B incorrect in this multiple choice?
Because atwoods are now more complex. The tensions (T1) and (T2) are no longer equal because there is a mass component there must now be a torque component.
Why is the correct answer "V" or "X" only for magnitude of force and mass In this balance problem?
Because for W you never put a force at the fulcrum since r= zero.
Why does freezing the can not lead to an effective strategy?
Because it means you have to rotate everything frozen which means more rotational kinetic and bigger moment of inertia (harder to rotate)
If each object has the same mass then why do some go down the ramp faster?
Because they have different mass distributions
As radias gets bigger what happens to moment of inertia?
Bigger moment of inertia and therefore harder to rotate
What does it mean if two objects are on the same rotating disk at different locations?
Both objects are rotating through the same number of radians in the same amount of time. Therefore they have the same w (angular velocity!)
Circumference Formula
C = 2πr
A wheel rolls without slipping. Which is the correct velocity vector for point P on this wheel?
C has the combination of A and B because it has both translational and rotational motion
Using the wheel scenario in a FRQ which case has the greatest speed?
Case 1 because there is no friction is ice so it has a larger acceleration and net force . Case 2 has friction
If an object is rotating then something must have
Cause it to change its inertia (torque) (turning effect)
Don't exclude
Coefficients when interpretation data Sometimes we can't calculate the known density of the mass.
What do energy bars connect with?
Coefficients!! Hoop must have 1/2 in front of it cause it has half and half of each kinetic.
Unit 7 real life applications
Collisions in rotations, finding velocity of objects rolling on ramp or spinning, objects in elliptical orbit, friction!!
A planet orbits a sun in an elliptical orbit, as shown. Which principles of physics most clearly and directly explain why the speed of the planet is the same at positions A and B? Select TWO answers.
Conservation of energy & conservation of Angular Momentum
Torque Equilibrium Equation
Counter clock wise torque = clock wise torque
Ladybug FRQ B The ladybug A walks in a circular motion with the disk. Does the angular momentum of the disk increase, decrease or stay the same
Decrease because the ladybug is external torque on the disk system. So the angular momentum of the ladybug increases and in order for angular momentum to be conserved the angular momentum of the disk alone must decrease.
What if we half the radias for student/ merrygoround?
Decrease but not cut in half because r^2.
A solid disk and a hollow hoop are poised at the top of the same ramp. All objects have the same mass and radias. If they are released simultaneously, which arrives at the bottom of the ramp first?
Disk
rotational inertia also depends on
Distribution of mass
Coefficient means it depends on
Distribution of mass. Largest coefficient of 1 = hoop was last place down the ramp: Smallest coefficient of sphere 2/5 was first place down the ramp:
How would the angular momentum change if we doubled the rotational inertia?
Double We would be doubling the variable I in the angular momentum equation
Advise for energy problems
Draw an energy bar chart
Conservation of energy for rotating objects
EI = EF (energy initial = energy final) Initial (GPE) = translational KE final+ ROTATIONAL KE final
Translational motion + rotational motion
Equals rolling
Does F1 or F2 apply the larger torque and why?
F2 does because it is farther away from the axis of rotation. Meaning a greater "r" value
Diagram of student jumping on merry go round (use vector arrows)
Find angular speed
How in the world is this FRQ part C correct?
Focus on square roots Don't pay attention to weather the numbers are correct. Focus on the variables. If time goes up then the variable better be in the numerator if it also goes up. If time goes down then the variable better be in the denominator going up.
Is linear momentum conserved in this FRQ.
For B and C because there is no pin, The pin is an external force so A and D are not
Cool physics conversion
For angular velocity. Units for w are "rad/sec" So take revolutions and convert it to radians than divide by 7.6 seconds!
What is the free body diagram of the wheel scenario?
Force of gravity from center of mass of the wheel to straight down. Then force normal which is diagonal because it must start from the surface and goes beyond the halfway point of the wheel center of mass at an angle. Force friction opposing the motion of the wheel.
Rotational motion diagram
Four arrows with up and down. Going right tangential velocity is VT= w * r And left it's -w* r
Why is the bottom of rolling motion the wheel is zero?
Friction allows it to roll
A wheel is on a ramp. It doesn't rotate it just slides... we then change the surface and it starts to rotate... what caused it to rotate?
Friction! Which applies the torque
disk cylinder energy bar chart
GPE INITIAL = 3/4 Translational kinetic and 1/4 rotational kinetic
Solid sphere energy bar chart
GPE INITIAL = 4/5 kinetic translational & 1/5 rotational kinetic energy
hoop energy bar chart
GPE INITIAL = half and half of kinetic translational and rotational kinetic Equal to each other
Hallow sphere energy bar chart
GPE INITIAL= 3/5 translational kinetic & 2/5 rotational kinetic (between hoop and disk)
Objects translating
Go in a straight line but still can have angular momentum about some point
Force of friction in wheel scenario
Goes against the motion from center of bottom
Motion detector creates
Graphs Used to find velocity or position for an object
Cube of ice energy bar chart
Gravitational = all translational kinetic No rotational anything
What type of energies are involved in the ramp scenario?
Gravitational potential because= if all objects have the same mass and are at the same height they will have the same GPE. There is kinetic energy KE= 0.5 *m* v^2 however.... We broke physics because at the bottom of the ramp the objects don't have the same velocity. Or rotational kinetic energy
More moment of inertia means
Harder to rotate Therefore smaller angular acceleration
Moment of inertia variable
I
Point mass equation
I = M * R^2
conservation of angular momentum example in experimental design FRQ
On the right side of the equation we need I of the wheel and I of the hoop
One bike wheel rotation
One revolution (circumference)
MC Practice 1 correct or incorrect and why?
Incorrect. I said A and the answer was C C is correct because there is no external torque from the student pulling the string down so angular momentum is conserved. And then as the radius decreases the system will spin faster thus increasing KE. Think dumbbells moving inward. KE increases because there is work done by the force. 1/3= 9 times less rotational inertia Square 3
As student moves towards center what happens to angular velocity
Increases
To help the man across the narrow zip line he carried a steel pole that's heavy which...
Increases his moment of inertia (harder to rotate due to increase in mass), which results in a smaller angular acceleration giving him more time to balance.
As radius gets smaller what happens to angular velocity?
Increases!
Omega vs time graph
Is a velocity vs time graph To find the angular displacement between 0-30 seconds we just need to calculate the area by breaking down the shape into two triangles and one narrow rectangle. To get 52.5 radians.
How does a treadmill and stationary bike know how far we go?
It calculates the number of radians the wheel or belt rotated and multiples it by the radias. Giving us distance.
As rotating dumbbell person pulls dumbells towards their body what happens to the angular speed?
It increases because rotational inertia gets smaller. The farther away from center= slower
What is true about F3?
It is not a torque. It is just pulling on it. So it won't cause rotational motion since it's just Parrel.
What does 212° mean ?
It will go 21 spaces past 15 From 0.59 revolution
Suppose that the rotational inertia was twice as great... how would the new change in angular momentum compare?
It would be the same because we are using the L= torque * time equation There is no moment of inertia
translational kinetic energy
It's the kinetic energy of an object that moves from one point to another with some velocity.
The linear distance (arc length) a wheel moves is proportional to
Its radius
The kinematic equations only hold true if there are no
Jerks (change in acceleration) so acceleration must be constant
rotational Kinetic Energy Formula
KE = 0.5 * I * W^2
Translational Kinetic Energy Formula
KE = 0.5mv^2
How can a moving point mass have an angular momentum?
L= m * v * r or L= I * W * r
angular momentum equation
L=I* w Angular momentum = moment of inertia * angular velocity
The solid sphere we put in the Atwood has
Less moment of inertia. This means it will be easier to rotate. (Less resistant to rotating) so the acceleration of the system will be higher with the sphere.
In terms of rotational kinetic energy what does having a smaller I mean
Less rotational kinetic energy
How does the beetles linear velocity (vb) compare to the ladybug's linear velocity (vl)?
Less then because the radias is smaller.
Conservation of angular momentum equation
Li1+ Li2 = Lf1 + Lf2
Arc length is
Linear displacement
What is the relationship between linear velocity and angular velocity?
Linear velocity is proportional to both angular velocity and distance from gen axis of rotation.
Factors that influence inertia
Mass, Shape, Distribution of Mass.
To get Can down fastest we want to
Max translational KE and Minimize rotational KE
What does increasing moment of inertia mean
More resistive to rotation
To win derby we want
More translational because rotating takes more energy.
units for torque
NM (Newton meters)
Accelerations are caused by
Net torques and net forces
Is kinetic energy conserved in this FRQ?
No because all these collisions are inelastic
Force parrell times r equals
No torque
Force and radias parrell means
No torque angular momentum is conserved
If the momentum is constant does that mean the energy must be as well?
Not always! Think ice skater putting arms in she speeds up angular velocity increases so kinetic energy increases internally but not momentum.
Angular momentum is conserved in all collisions including those with
Objects that move linearly
When dividing by 2PIE always use
Parenthesis!! Or the calculator will mess up
What is a key for this unit?
Pay close attention to rounding.
Experimental practice wheel
Quantities to find I Scale for mass Ruler for radius Period of rotation stopwatch
What is more impactful mass or radias for moment of inertia? A or D debate?
Radias is more impactful because point mass equation says r is squared in the equation (I = M * R^2) D has a larger moment of inertia than A
How do we conserve angular momentum?
Reduce angular velocity to account for equal angular momentum of the other object
Examples of symbols
Rh = radius of hoop Mh= mass of hoop T= period
In order for conservation of momentum to be true the person must
Rotate the other way the ball is thrown (explosion)
As long as the object is rotating with ??? The equations will hold true.
Rotating with a constant angular acceleration (no jerks!)
What two things does angular momentum depend on?
Rotational inertia & angular velocity
A windmill has just
Rotational kinetic energy
Any rotating object will have
Rotational kinetic energy or translational depending on the object
Alpha vs torque graph
Slope = 1/2 I
Angular momentum vs time graph
Slope= net torque (rise/run)
We know our object is a solid cylinder in experimental design because we 5 is half of ten
So it must be a solid cylinder
Order of the objects that hit the bottom of the ramp from first to last place
Solid sphere first, disk second, hollow sphere third, and then hoop last Frictionless ice cube first
As a student moves towards the center of a merrygoround what happens to angular momentum?
Stays the same (no external torque)
What happens if spinning dumbbell person throws the dumbells in terms of angular momentum?
Stays the same because there is no external torque.
What happens if we add mass to this atwood system?
The acceleration will be less
Which object if either reaches the bottom of the ramp with greater speed?
The block because it doesn't experience friction like the wheel: the wheel has more rotational motion. Both have the same amount of potential energy in the beginning. However, at the bottom of the ramp the blocks energy is all translational kinetic while the wheel has both translational and kinetic energy so it can't be moving as fast as the block.
Which object in the ramp scenario took the most time to get to the bottom?
The hoop All the mass is located the same distance from the axis of rotation.
If we have a disk and a hoop of the same radius and mass.... Traveling with the same velocity which will reach a greater height on the ramp?
The hoop will because it has more rotational kinetic energy than the disk. And both objects have the same translational kinetic. Objects are moving up ramp!
A solid disk and a hoop are simultaneously released from rest at the top of an incline and roll down without slipping. Which object reaches the bottom first?
The disk arrives first.
What is linear distance dependent on?
The distance from the center. So the farther from the center means greater liner speed.
The key to rotational kinetic energy is that
The energy of an object will have angular velocity and rotating from one point to another.
Notice that the longer the torque is applied...
The faster the wheel moves and therefore it has greater angular momentum.
Right again for FRQ PART F!!
The graph in segment t is inconsistent again because with friction the object is not at rest... it is instead decreasing speed at a constant speed so itt will go down linearly.
The greater the rotational inertia
The greater torque needed for same angular acceleration
What has a bigger Moment of inertia hoop or disk!
The hoop
Does the beetle or ladybug have a greater centripetal acceleration ac?
The ladybug does because it is farther from the center. The radias of the beetle is 1/3 the radias of the ladybug so the beetle is 1/3 the ac of the ladybug.
smaller rotational inertia
The less rotational kinetic energy the object will have at the bottom. And the more translational kinetic energy it will have at the bottom.
Which Can will win down the ramp frozen or liquid?
The liquid because it has the water on the bottom. Less to rotate just the can. Mass stays on bottom.
The larger the rotational inertia... (use energy)
The more rotational kinetic energy the object will have at the bottom. And the less translational kinetic energy it will have at the bottom.
In this Unit along with unit 5 for conservation of angular momentum we cannot mix up
The motions! It's either angular momentum all the way through or linear momentum.
If a rod rotates in 8.3 seconds versus another object at 11.7 what does the time difference mean?
The rotational inertia of the rod is less than the other object
In kinematics of graphs in rotational motion the properties are
The same!
Where is rotational inertia in this graph?
The slope is the reciprocal of rotational inertia I= 5 kg * m^2 Rise over run = 1/5 Reciprocal = 5 Which is our moment of inertia.
Which object in the ramp scenario took the least amount of time to hit the bottom of the ramp?
The solid sphere Mass is spread throughout.
Inertia
The tendency of an object to resist a change in motion
If a hoop is replaced with a disk... what must happen to the torque of the disk to get it to be equal to the torque of the original hoop for the same rotations rate?
The torque of the disk must be less than the torque of the hoop. Since disk has smaller moment of inertia needing less torque.
As an object's mass increases, what happens to its moment of inertia?
it increases Depending on mass distribution as well
What else could account for a smaller angular acceleration besides mass?
There is a tension force at play. The mass is not in free fall. That means the force applying the torque is less than that of hanging mass. The group has too high of torque values. So angular acceleration would be lower.
If a gymnast opens their arms what happens?
They will slow down and come to a stop.
If a figure skater pulls arms In what will happen?
They will spin faster
Slope of velocity vs time
This slope equals acceleration So one line starts at zero (rest)
Why do we want the period? Experimental design
To get the angular velocity
Torque net equation
Torque net = moment of inertia * angular acceleration Tnet = I * a (alpha)
Example of change in angular momentum
Units are 0.08 kg * m^2 / sec
How do we derive an equation in terms of I2 for I1?
Use LI= LF I1W1= L2W2 Don't forget that 2Pies cancel out in step 4
Because the student moves in a straight line
Use point mass and linear velocity variables (r)
How can we calculate the torque of an object experimental design to then find our moment of inertia
Use torque net equation. Torque= f * r Then torque net = moment of inertia * angular acceleration We apply a force and use a force senor Then we use an angular motion sensor that can collect angular position, angular velocity, and angular acceleration. We will use a meter stick to measure the radias from central axis to edge.
How do we alalyze data to to determine rotational inertia of the wheel?
Use unit 6 period qualities
Angular moment is a
Vector quantity
Why is A correct?
Vertical intercept represents the force which means torque. So it is more significant than the slope. Because we are not sure what the slope means.
velocity final equation
Vf=Vi+at Linear graph
Velocity squared equation
Vf^2 = vi^2 + 2*a* change in X
shape constant
k
How do we find out what angular displacement actually means?
We convert theta into revolutions 72.82 rad from theta divided by 2PIE tells us revolutions are 11.59 It completes 11 full revolution and more than half of another.
Uniform circular motion
We move at a constant speed, but not a constant velocity because of centripetal acceleration.
What is the key to solving this multi moment of inertia problem?
We must calculate each point for Moment of inertia (4 seats / points total), some have children, others are just the seat. So different masses multiplied by the same radias
FRQ PART A WHERE SHOULD PUCK HIT to get most angular speed?
We want a big torque Farthest from hinges (bigger r)
Procedure means
What data we measure and what equipment we use
Rotational motion
When an object rotates around a fixed axis
conservation of angular momentum
When no external torque acts on an object or a system of objects, no change of angular momentum can occur. Hence, the angular momentum before an event involving only internal torques or no torques is equal to the angular momentum after the event.
In what type of situations would you expect angular momentum to be conserved?
When there is any type of collision, or when there is a shape or lass distribution of the object.
WOT equation in action
With one revolution we know that our angular displacement is 2PIE
How do we find velocity of a ball thrown before it moves together with a disk?
Work shown (return)
While the ladybug is walking toward the center of the disk, does it exert a torque on this disk?
Yes Because the the angular momentum of the disk changes because the angular speed of the disk increases and its rotational inertia does not change. The change in angular momentum is directly proportional to the torque. There must be a torque applied because the angular momentum changed.
Does increasing mass increase omega in this FRQ according to our reasoning from part A?
Yes For indicating the equation shows w increasing with X. According to the equation bigger x will produce a bigger angular speed.
if the student jumped off the merrygoround would angular momentum be conserved?
Yes because merrygoround would stationary just like the beginning and no net external torque was applied.
Is angular momentum constant for this FRQ?
Yes there are no extrernal torques
Will decreasing the radius (making the bike wheel smaller) affect how the wheel rotates?
Yes. It went farther! 650mm for one rotation
FRQ PART G significance
You were right about the acceleration going down after "T" because of friction. But we must also remember that it will go down in the negative direction because it is slowing down due to friction.
Torque
a force that causes rotation by turning or twisting
Conservation of energy
a principle stating that energy cannot be created or destroyed, but can be altered from one form to another.
Hoop
a ring or round band
centripetal acceleration equation
ac = v^2/r
Second equation for centripetal acceleration
ac= w^2 * r
Rotational inertia
also known as the moment of inertia
Alpha
angular acceleration
Rotating objects have
angular momentum
Omega
angular speed
If the student ran at the merrygoround with twice the velocity what would happen to the velocity of the merrygoround based off of our equation?
angular velocity of merrygoround would also double
The units of angular momentum are __________.
kg*m2/s
An ice skater is spinning about a vertical axis with arms fully extended. If the arms are pulled in closer to the body, in which of the following ways are the angular momentum and kinetic energy of the skater affected?
b) angular momentum- remains constant kinetic energy- increases
Moment of inertia units
kgm^2 (kilograms * meters squared) !!!
Without external torques an object __________.
can change its own angular velocity by rearranging its mass distribution
Angular impulse
change in angular momentum due to a net torque
acceleration vs time graph
horizontal line shows constant acceleration area = change in velocity
θ
how many radians the wheel rotates through.
Angular momentum of point mass moving in a linear fashion
mvr
clockwise rotation
negative left
Another way of saying net torque = moment of inertia TIMES angular acceleration
net torque = moment of inertia TIMES angular velocity/ change in time
counter clockwise rotation
positive Right
units for angular acceleration
rad/s^2
units for angular velocity
rad/sec (radians per second)
r in rotational torque
radical distance
r
radius
angular velocity
rate of change of angular displacement
θ
represents angular displacement
Rotating objects still follow
the law of conservation of energy as long as there is no external force
Moment of inertia is
the rotational equivalent of mass
Angular momentum is conserved when
there are no external torques acting on a system
Torque equation including perpendicular component
torque = r * f * sin (theta)
if an object is rolling without slipping, how does its linear speed compare to its rotational speed?
v = rω
Velocity in centripetal acceleration (v) is equal to
v= w*r
angular velocity equation
wf= wi + alpha * time
Angular velocity squared equation
wf^2 = wi^2 + 2* alpha * change in θ
Linear displacement symbol
x
position equation
x(t) = xi+ vi * t + 0.5* a* t^2 Curvy graph
How distance traveled is determined
x= θr
change of angular momentum
ΔL=τt change in angular momentum = torque * time
angular acceleration equation
α= Δω/Δt
Theta symbol
θ
angular displacement equation
θ= θi + wi * t + 0.5 * alpha * t^2
angular velocity equation
ω = θ/t
