Chapter 10
Angular Acceleration (α(avg))=
(α(avg))=Δω/Δt (α(avg))=change in angular velocity/time
What does torque depend on?
-Magnitude of the force -Moment arm/lever arm --The direction in which it acts --The point at which it is applied on the object
What are the two conditions for a center of mass object?
-mass distributions are uniform -it is a simple geometric form/shape
what is a physical quantity
A quantity that can be measured (has magnitude but not direction)
Rotational kinematics (just like linear kinematics) is descriptive and [does/does] not represent laws of nature. With kinematics, we can describe many things to great precision but kinematics [does/does not] consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity [with/without] any consideration of its cause.
Answer 1 & 2: does not Answer 3: without
The units of angular acceleration are .............................. . If ω increases, then α is [positive/negative]. If ω decreases, then α is [positive/negative].
Answer 1: (rad/s)/s , or rad/s2 Answer 2: positive Answer 3: negative
.............................. acceleration refers to changes in the magnitude of velocity but not its direction. We know that in circular motion ..................................... refers to changes in the direction of the velocity but not its magnitude.
Answer 1: Linear or tangential Answer 2: centripetal acceleration, ac
Given the following equation α=Δω/Δt, if w is increasing the vector α and the vector w are ....................... and Δω is .................. Answer 1 options: a. going in the same direction b. going in opposite directions Answer 2 options: a. positive b. negative
Answer 1: a. going in the same direction Answer 2: a. positive
If the torque you exert is greater than opposing torques, then the rotation [accelerates/decelerates], and angular momentum [increases/decreases]. The [greater/less] the net torque, the [more rapid/less rapid] the increase in L . The relationship between torque and angular momentum is netτ=ΔL/Δt. This expression is exactly analogous to the relationship between force and linear momentum, ....................... . The equation netτ=ΔL/Δt is very fundamental and broadly applicable. It is, in fact, the rotational form of Newton's ................... law.
Answer 1: accelerates Answer 2: increases. Answer 3: greater Answer 4: more rapid Answer 5: F=Δp/Δt Answer 6: second
Given the following equation α=Δω/Δt, if w is decreasing the vector α and the vector w are ....................... and Δω is .................. Answer 1 options: a. going in the same direction b. going in opposite directions Answer 2 options: a. positive b. negative
Answer 1: b. going in opposite directions Answer 2: b. negative
Centripetal acceleration ac occurs as the [magnitude/direction] of velocity changes; it is [parallel/perpendicular] to the circular motion. Centripetal and tangential acceleration are thus [parallel/perpendicular] to each other.
Answer 1: direction Answer 2 & 3: perpendicular
linear acceleration and angular acceleration are [directly/indirectly] proportional. The greater the angular acceleration is, the [larger/smaller] the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car's drive wheels, the [greater/less] the acceleration of the car. The radius also matters. For example, the smaller a wheel, the [larger/smaller] its linear acceleration for a given angular acceleration α .
Answer 1: directly Answer 2: smaller Answer 3: greater Answer 4: smaller
Angular velocity [is/is not] constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer's hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration, in which ω [remains constant/changes]. The [faster/slower] the change occurs, the greater the angular acceleration.
Answer 1: is not Answer 2: changes Answer 3: faster
I is analogous to ..................... in translational motion. Because of the distance r , the moment of inertia for any object depends on .............................
Answer 1: m Answer 2: the chosen axis.
• L=Iω. This equation is an analog to the definition of linear momentum as ............................. Units for linear momentum are .................... while units for angular momentum are .................. As we would expect, an object that has a large moment of inertia I , such as Earth, has a very [large/small] angular momentum. An object that has a [large/small] angular velocity ω , such as a centrifuge, also has a rather large angular momentum.
Answer 1: p=mv Answer 2: kg⋅m/s Answer 3: kg⋅m^2/s Answer 4 & 5: large
An object undergoing circular motion experiences centripetal acceleration Thus, at and ac are [parallel/perpendicular] and [independent of/dependent on] one another. Tangential acceleration at is [directly/indirectly] related to the angular acceleration α and is linked to an increase or decrease in the velocity, but not its direction.
Answer 1: perpendicular Answer 2: independent Answer 3: directly
According to the sign convention, the counter clockwise direction is considered as ........................ direction and clockwise direction as .....................................
Answer 1: positive Answer 2: negative
The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the [larger/smaller] is the angular acceleration. But there is an additional twist. The moment of inertia depends not only on the mass of an object, but also on....................................................
Answer 1: smaller Answer 2: its distribution of mass relative to the axis around which it rotates.
How Thick Is the Soup? Or Why Don't All Objects Roll Downhill at the Same Rate? -One of the quality controls in a tomato soup factory consists of rolling filled cans down a ramp. If they roll too fast, the soup is too [thin/thick]. Why should cans of identical size and mass roll down an incline at different rates? And why should the thickest soup roll the slowest? -The easiest way to answer these questions is to consider ....................... Suppose each can starts down the ramp from rest. Each can starting from rest means each starts with the same gravitational potential energy PEgrav , which is converted entirely to KE , provided each rolls without slipping. KE , however, can take the form of ...................... or .......................... , and total KE is the sum of the two. If a can rolls down a ramp, it puts part of its energy into rotation, leaving less for translation. Thus, the can goes [faster/slower] than it would if it slid down. Furthermore, the thin soup [does/does not] rotate, whereas the thick soup [does/does not], because it sticks to the can. The thick soup thus puts [more/less] of the can's original gravitational potential energy into rotation than the thin soup, and the can rolls more [quickly/slowly]
Answer 1: thin Answer 2: energy Answer 3: KEtrans Answer 4: KErot Answer 5: slower Answer 6: does not Answer 7: does Answer 8: more Answer 9: slowly
In statics, the net torque is ..........................., and there is no angular acceleration. In rotational motion, net torque is the cause of.............................., exactly as in Newton's second law of motion for rotation.
Answer 1: zero Answer 2: angular acceleration
Total acceleration=
At+Ac
A point inside the object/system of particles where all of the mass can be though to be concentrated & net force acted on it.
Center of Mass, CM
deals with the study of motion of objects taking into account the factors which cause motion (based on concept of force)
Dynamics
True or False? Rigid objects are point like
FALSE
true or false? the angle of the force applied makes no difference
FALSE
true or false? When considering the rotational kinematics of rigid objects speed remains constant
FALSE Speed does not have to be constant.
True or False? All points in a rigid object do not rotate with the same angular velocity
False
True or False? the larger the torque is, the smaller the angular acceleration is.
False the larger the torque is, the larger the angular acceleration is.
For rotational motion, kinematics answers the question..................
How is it rotating?
deals with study of motion of material objects without taking into account the factors which causes motion (i.e. nature of force, nature of body)
Kinematics
Rigid objects are non-deformable, meaning?
Solid: external forces will be consistent. Relative positions of all particles inside remain fixed.
the turning effectiveness of a force
Torque
True or False? Hollow objects can still have a center of mass
True Hollow, can still be considered the center of the mass because the kinematics and dynamics do not change because that is where all the mass acts upon.
For rotational motion, dynamics answers the question ......................
Why is it rotating?
V=rw r is the radius rb>ra so, Vb................va Answer options: a. > b. < c. =
a. >
For only translational motion of the CM, a diver would a. jump straight upward and not rotate when falling into the water b. spin while falling into the water.
a. jump straight upward and not rotate when falling into the water
If the rotation is in the counter-clockwise direction what will w be? a. positive b. negative
a. positive
If the rotation is in the counter-clockwise direction what will Δθ be? a. positive b. negative
a. positive
the rate of change of angular velocity with time
angular acceleration
Rotational kinematics considers
angular displacement angular velocity angular acceleration
Average angular velocity (Wavg)=
angular displacement/time Wavg=Δθ/Δt -How far does it turn for that amount of time
The collision factor considered by rotational kinematics is
angular momentum
the product of moment of inertia and angular velocity
angular momentum
rate of change of angular displacement
angular velocity
θ=
arc length/radius=l/r
If the rotation is in the clockwise direction what will w be? a. positive b. negative
b. negative
If the rotation is in the clockwise direction what will Δθ be? a. positive b. negative
b. negative
For only rotational motion of the CM, a diver would a. jump straight upward and not rotate when falling into the water b. spin while falling into the water.
b. spin while falling into the water.
the difference between final and initial values of angular velocity
change in angular velocity
Instantaneous angular acceleration
change in angular velocity at one specific instant in time
true or false? Rotation of a single pint is not enough to study
false
true or false? the smaller the radius the more rotation
false
True or False? Torque and force are the exact same thing
false, they're similar but not exactly the same. Force is the energy that makes something move and torque is the effectiveness of the force used to make something move
Instantaneous angular velocity
how fast something is rotating at a specific instant in time
describes the relationships among rotation angle, angular velocity, angular acceleration, and time
kinematics of rotational motion
angular momentum is conserved, i.e., the initial angular momentum is equal to the final angular momentum when no external torque is applied to the system
law of conservation of angular momentum
the study of motion
mechanics
What quality that torque depends on is different from force?
moment arm/lever arm
mass times the square of perpendicular distance from the rotation axis; for a point mass, it is I=mr2 and, because any object can be built up from a collection of point masses, this relationship is the basis for all other moments of inertia
moment of inertia
the reference line in space is always
on the x axis
What is an axis of rotation?
purely rotational motion about a fixed axis.
What is the SI unit of w?
rad/s
What is the SI unit for angular acceleration?
rad/s^2
What is the units of θ?
radian "rad" -calculator must be in radian mode
direction of angular velocity ω and angular momentum L in which the thumb of your right hand points when you curl your fingers in the direction of the disk's rotation
right-hand rule
Torque is associated with ..................motion
rotational
extended shape of objects needs to be considered (rigid objects) in ...............motion
rotational
resistance to change of rotation. The more rotational inertia an object has, the harder it is to rotate
rotational inertia
The energy consideration of rotational kinematics is
rotational kinetic energy
the kinetic energy due to the rotation of an object. This is part of its total kinetic energy
rotational kinetic energy
Linear acceleration is also called................
tangential acceleration
the acceleration in a direction tangent to the circle at the point of interest in circular motion
tangential acceleration
linear acceleration is called
tangential acceleration (at)
Rigid object are extended objects, meaning?
the whole shape must be considered
Rotational dynamics considers
torque and force
objects are treated as point-like (motion of the CM) in ............motion
translation motion
Force is associated with ..................motion
translational
True or False? All points in a rigid object rotate with the same angular acceleration
true
True or False? Each point moves in a circle about the rotation axis in a spinning top where the axis of rotation is perpendicular to the disk and passes through "o".
true
True or False? For At, the larger the radius or the farther away from the center, the larger it will be
true
True or False? In translational motion the values for displacment, velocity, and acceleration change throughout the object, but they remain constant throughout the object in rotational motion
true
True or False? V will not be the same for all points inside a rigid body
true
True or False? W will be the same for all points inside a rigid body
true
True or False? there is translational motion even for something spinning in place, as the following example illustrates
true
True or false? When considering rotation about the z axis, the rotating object doesn't actually go in the positive Z directions, it's just a convention
true
true or false? All points in a rigid object rotate by the same angular displacement
true
true or false? Extended shape is the main difference between translational and rotational motion
true
true or false? the bigger the force, the more rotation
true
True or False? Any point in an object with an expanded shape, there is a point where all of the mass can be though to be concentrated.
true center of mass, CM
true or false? a baseball bat can have a center of mass
true The left side is longer but thinner. the right side is shorter but more dense so you know more mass is toward the right and the center of mass is shifted toward the right. Sometimes will have to calculate where it is, but that requires calculus. We will only approximate
Ac=
v^2/r
What kind of quantity is torque?
vector quantity
What is the symbol for angular velocity?
w
if one or more external forces act upon a rigid object, causing its kinetic energy to change from KE1 to KE2, then the work W done by the net force is equal to the change in kinetic energy
work-energy theorem
In terms of rotational motion v=
Δl/Δt=rΔθ/Δt=rw
At=
Δv/Δt=rα How fast velocity is changing in the tangential direction
Δθ=
θ2-θ1 -measured in "rad"
