General Physics 1: Quarter 2 - Module 1
General Physics 1: Quarter 2 - Module 1
General Physics 1: Quarter 2 - Module 1
General Physics 1
Quarter 2 - Module 1
Rotational Equilibrium and Rotational Dynamics
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Reviewers:
General Physics
Quarter 2 - Module 1
Rotational Equilibrium and
Rotational Dynamics
What I Know...................................................................................................................................iii
Summary......................................................................................................................... 26
Assessment: (Post-Test)................................................................................................. 28
Key to Answers................................................................................................................ 30
References 31
Module 1
Rotational Equilibrium and Rotational
Dynamics
What This Module is About
The first part of this module will discuss the kinematics of rotational motion as
described in its angular position, speed and acceleration. The second part will
discuss its dynamics and explain the forces that causes objects to rotate. In this part,
you will understand the Physics of simple events observed daily such as the motion
of opening doors, the concept behind see-saw; the motion of ice-skaters and many
more.
1. Calculate the moment of inertia about a given axis of single-object and multiple object
systems STEM_GP12REDIIa-1
2. Calculate magnitude and direction of torque using the definition of torque as a cross
product STEM_GP12REDIIa-3
3. Describe rotational quantities using vectors STEM_GP12REDIIa-4
4. Determine whether a system is in static equilibrium or not STEM_GP12REDIIa-5
5. Apply the rotational kinematic relations for systems with constant angular accelerations
STEM_GP12REDIIa-6
6. Solve static equilibrium problems in contexts such as, but not limited to, seesaws,
mobiles, cable-hinge-strut system, leaning ladders, and weighing a heavy suitcase using
a small bathroom scale STEM_GP12REDIIa-8
7. Determine angular momentum of different systems STEM_GP12REDIIa-9
8. Apply the torque-angular momentum relation STEM_GP12REDIIa-1
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
• Take your time reading the lessons carefully.
• Follow the directions and/or instructions in the activities and exercises diligently.
• Answer all the given tests and exercises.
Multiple Choice. Answer the question that follows. Choose the best answer from the
given choices.
1. Which of the following statements show the properties of angular displacement and linear
displacement?
A. θ, for points on a rotating object depends on their distance from the axis.
B. θ, for points on a rotating object does not depends on their distance from the axis.
C. The displacement, for point on a rotating object, depends on their distance from the
axis of rotation.
D. The displacement, for point on a rotating object, does not depends on their distance
from the axis of rotation
2. What is the linear speed of a child on a merry-go-round of radius 3.0 m that has an
angular velocity of 4.0 rd/s?
A.) 0.75 m/s B.) 10 m/s C.) 12 m/s D.) 13.0 m/s
3. What is the angular velocity of an object traveling in a circle of radius 0.75m with a linear
speed of 3.5 m/s?
A.) 4.3 rd/s B.) 4.8 rd/s C.) 4.9 rd/s D.) 4.7 rd/s
4. What is the angular acceleration of a ball that starts at rest and increases its angular
velocity uniformly to 5 rd/s?
A.) 8.0 rd/s2 B.) 2 rd/s2 C.) 0.5 rd/s2 D. 3 rd/s2
5. What is the angular velocity of a ball that starts at rest and rolls for 5 seconds with a
constant angular acceleration of 20 rd/s2?
A.) 4 rd/s B.) 10 rd/s C.) 100 rd/s D.) 7 rd/s
6. If no external torque acts on a body, its angular velocity remains conserved.
A) True B) False
7. The easiest way to open a heavy door is by applying the force
A) Near the hinges B) In the middle of the door
C) At the edge of the door far from the hinges D) At the top of the door
8. Does a bridge anchored resting on two pillars have any torque?
A) No, it isn't moving B) Yes, but it is at equilibrium
C) Yes, but it will soon break because of the torque D) No, Bridges can't have torque
9. When an object is experiencing a net torque
A) it is in dynamic equilibrium. B) it is in static equilibrium.
C) it is rotating. D) it is translating.
10. A rusty bolt is hard to get turned. What could be done to help get the bolt turned?
A) use a long-arm lever B) decrease the force
C) apply the force at a 30 degree angle D) use a short-arm lever
Lesson
1 ROTATIONAL KINEMATICS
What’s In
Have you ever watched a Ferris wheel as it turns? How do you feel? Did you
ever wonder how it moves? Will you still ride it if it doesn’t turn? This is why
rotational motion is a very important motion. It is important to know how this motion
affects the movement of a certain body.
What’s New
What Is It
Rotational Kinematics
Kinematics is the description of motion. It is concerned with the description of
motion without regard to force or mass. But what exactly is rotational kinematics?
From the word, you can describe that it’s all about any object that can rotate or spin.
It’s different from linear motion when object simply moves forward. The kinematics of
rotational motion describes the relationships among rotation angle (θ), angular
velocity (ω), angular acceleration ( α) , and time (t). You will find that translational
kinematic quantities, such as displacement, velocity, and acceleration have direct
analogs in rotational motion.
Axis of Rotation
In activity 1.1, you have listed some types of rotating objects and their
importance to society right? Everything that you have listed are all rotating about a
line somewhere within the object called the axis of rotation. We are also going to
assume that all these objects are rigid bodies, that is, they keep their shape and are
not deformed in any way by their motion. Look at Figure 1 below. It shows the wheel
and axle of a bike. Is the axle (axis of rotation) part of the wheel (rigid body)? The
answer is NO. If you were to spin the wheel around its center, the axis of rotation
(axle) would be pointing perpendicular to the motion of the wheel.
2
Angular Displacement
We will now define the angle of rotation (θ) as the ratio of the arc length (s) to
the radius (r) of the circle. We call this angle of rotation (θ) the angular
displacement. We denote angular displacement as Θ (theta). In symbol,
3
Angular displacement is unitless since it is the ratio of two distances but, we
will say that the angular displacement is measured in radians. We know degrees,
and we know that when a point on a circle rotates and comes back to the same
point, it has performed one revolution; let us say from point A, and rotate until we
come back to point A.
Refer to Figure 2 again, what distance (s) was covered? How many degrees
were swept by this full rotation? The point moved around the entire circumference,
so it traveled 2πr while an angle of 3600 was swept through. Using the angular
displacement definition:
Sample Problems
1. ) An object travels around a circle10.0 full turns in 2.5 seconds. Calculate (a) the
angular displacement, θ in radians.
Given:
# of turns/complete rotations = 10 turns
Time = 2.5 seconds
Find: Angular displacement (θ) in radians
Solution:
Θ = 10.0 turns ( 6.28 rd / turn ) = 62.8 radians.
2. ) A girl goes around a circular track that has a diameter of 12 m. If she runs
around the entire track for a distance of 100 m, what is her angular displacement?
Given:
Diameter of the curved path = 12m ;
*Note that diameter = 2r therefore,
r= d/2 so,
r= 12m/2= 6m
Linear displacement, s = 100 m.
4
Find: Angular displacement θ
Solution:
Θ = s/r → θ = 100m / 6 m = 16.67 radians
Angular displacement can now be related to linear displacement. Working on
Kinematics problems with linear displacement was tackled in your previous lessons.
What other quantities played a key role in linear displacement?
Angular Velocity
In linear motion, velocity (v) is defined as the rate of change of the object's
position with respect to a frame of reference and time that is, v=∆ x /∆ t
while acceleration (a) is the rate of change of velocity. In symbol, we have:
a=∆ v /∆ t ; a=( v 2 − v 1 )/∆ t
In rotational motion, angular velocity (ω) is defined as the change in angular
displacement (θ) per unit of time (t). In symbol,
ω=∆ θ /∆ t
The symbol ω is pronounced "omega" is used to denote angular velocity.
Sample Problems
1.) If an object travels around a circle with an angular displacement of 70.8 radians
in 3.0 seconds, what is its average angular velocity ω in (rd/s)?
Answer
2.) A bicycle wheel with a radius of 0.28 m starts from rest and accelerates at a rate
of 3.5 rad/s2 for 8 s. What is its final angular velocity?
Answer
Given: r = 0.28 m; α = 3.5 rd/s2 t = 8 s
Find: ω =?
Solution: From the equation α = ω / t , we can have
ω = αt = 3.5 rd/s2 ( 8s ) = 28 rd/s
Angular Acceleration
If the angular velocity of the rotating object increases or decreases with time,
we say that the object experiences an angular acceleration, α. The angular
acceleration of a rotating object is the rate at which the angular velocity changes with
respect to time. It is the change in the angular velocity, divided by the change in
time. The average angular acceleration is the change in the angular velocity, divided
by the change in time. The angular acceleration is a vector that points in a direction
along the rotation axis. The magnitude of the angular acceleration is given by the
formula below. The unit of angular acceleration is radians/s 2.
6
In symbol,
Where:
α = angular acceleration, (radians/s2)
Δω = change in angular velocity (radians/s)
Δt = change in time (s)
ω1 = initial angular velocity (radians/s)
ω2= final angular velocity (radians/s)
t1 = initial time (s)
t2= final time (s).
All points in the object have the same angular acceleration. Every point on a
rotating has, at any instant a linear velocity (v) and a linear acceleration (a). Look at
the illustration in Figure 1.4 below, we can relate the linear quantities (v and a) to the
angular quantities (ω and α). Linear velocity and angular velocity are related since
v = rω
Where; v is the linear velocity,
r is the radius of the object, and
ω is the angular acceleration.
It is not only the point (we measure) move in that angular velocity. All points in
the object rotate with the same angular velocity. Every position in the object move
through the same time interval. Conventionally, object moving counterclockwise has
a value of positive (+) angular acceleration, while the one moving the clockwise
direction has negative (-) value.
Sample Problems
1. A disc in a DVD player starts from rest, and when the user presses “Play”, it
begins spinning..The disc is spins at 160 radians/s after 4.0 s. What was the
average angular acceleration of the disc?
Answer:
Given:
T1 = 0 T2= 4.0 s
ω1 = 0 ω2 =160 rd/s
Find:
Angular acceleration (α) =?
Solution:
Between the initial and final times, the average angular acceleration of the disc was
40.0 radians/s2.
2.) A car tire is turning at a rate of 5.0 rd / sec as it travels along the road. The
driver increases the car's speed, and as a result, each tire's angular speed
increases to 8.0 rd /sec in 6.0 sec. Find the angular acceleration of the tire.
Answer
Find: α=?
Solution:
Answer
Find: a) r b) ω2 c) θ
Solution:
ω2= ω1+ α Δt
Putting these definitions together, you observe a very strong parallel between
translational kinematic quantities and rotational kinematic quantities See Table 1.1
below.
The rotational kinematic equations (See table below) can be used the same
way you used the translational kinematic equations to solve problems. Once you
know three of the kinematic variables, you can always use the equations to solve for
the other two.
Source:https://www.aplusphysics.com/courses/honors/rotation/honors_rot_kinematics.html
10
What’s More
Activity 1.2 Matched Me Right
Match column A with column B according to their meaning. Write the letter of
your answer on the space provided before each number.
Column A Column B
(Meaning/Definition)
(Term/s)
___ 1. A measure of how angular velocity changes over time. A. Angular
___ 2. The imaginary or actual axis around which an object position
may rotate. B. Linear velocity
___ 3.It is the change in linear velocity divided by time. C. Axis of
rotation
___ 4. It is half of the circle’s circumference D. Tangential
___ 5. The orientation of a body or figure with respect to a Acceleration
specified reference position as expressed by the amount of E. Angular
rotation necessary to change from one orientation to the velocity
other about a specified axis. F. Kinematics
___ 6. The rate of rotation around an axis usually G. Angular
expressed in radian or revolutions per second or acceleration
per minute. H. Radian
___ 7. A property of matter by which it remains at rest or in I. Angular
uniform motion in the same straight line unless acted upon displacement
by some external force. J. Radius
___ 8. Branch of dynamics that deals with aspects of motion apart
from considerations of mass and force.
___ 9. It is the rate of change of the position of an object that is
traveling along a straight path.
___ 10. It is an angle whose corresponding arc in a circle is equal to
the radius of the circle
1. ) Mark bought a pizza of a radius of 0.5 m. A fly lands on the pizza and walks around the
edge for a distance of 80 cm. Calculate the angular displacement of the fly?
2. ) What is the angular velocity of an object traveling in a circle of radius 0.75 m with a
linear speed of 3.5 m/s?
3.) What is the angular acceleration of a ball that starts at rest and increases its angular
velocity uniformly to 5 rad/s in 10 s?
11
What I Can Do
1.) Angular acceleration does not change with radius, but tangential acceleration
does.
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3.) On a rotating carousel or merry-go-round, one child sits on a horse near the outer
edge and another child sits on a lion halfway out from the center. Which child has the
greater linear velocity? Which child has the greater angular velocity?
12
Lesson
2 ROTATIONAL DYNAMICS
In the previous lesson, you learned about the analogy of translational and
rotational motion. With this you were able to derived the basic variables necessary in
understanding this type of motion. Also, you learned the kinematics of rotating body without
taking into account the factors that causes its motion.
In this lesson you will understand rotational motion further through its dynamics;
that is how Torque, the force applied, causes a body to rotate. Also, in this lesson, you will
learn the conditions of Static Equilibrium; the Work done by the torque and the Angular
Momentum and their analogy to Newton’s Laws of Motion. This lesson will help you explore
and understand how simple events encountered and observed daily works. Specifically, you
are expected to learn the following:
13
What’s New
Investigating Torque!
Ease of Rotation
Situation (Rank the Forces from easiest to hardest)
1st 2nd 3rd
A. Opening a
Door
B. Removing a
Bolt using a
Wrench
C. Rotating A
Blade
From the results obtained and observed, deduce the relationship of the following:
14
A. TORQUE ( τ)
Have you ever wondered why doorknobs are situated at the opposite end of the
hinges and not near it? And why is it easier to use long-handled wrenches than the short-
handled one in removing bolts? How about doing an arm-wrestling with a longer-arm
person? What do you think would be your chances of winning?
This lesson will enlighten you on the simple physics behind these things. With the
understanding of Torque, you will be able to answer these questions.
Mathematically, ⃑ ⃑ ⃑
τ =r x F
whose magnitude is equal to τ =r ⊥ F
τ =rFsinθ
where r ⊥ =rsinθ
And θ is the angle between r and F
S.I. Unit: Nm
From the equation, we see that the effect of the Force on the motion of the rotating
body depends on three factors as follows:
Sample Problems:
Solution:
⃑
F=( 4 i−^ 3 ^j+ 5 k^ ) N
⃑ ^ 4 ^j− 2 k^ ) m
r =( 7 i+
| |
i^ ^j k^
⃑ ⃑ ⃑ ^ ( 35+ 8 ) ^j+ ( −21 −16 ) k^ =( 14 i^ − 43 ^j− 37 k^ ) Nm
τ =r x F= 7 4 −2 =( 20− 6 ) i−
4 −3 5
2. A crane has an arm length of 20 m inclined at 30º with the vertical. It carries a container
of mass of 2 ton suspended from the top end of the arm. Find the torque produced by the
gravitational force on the container about the
point where the arm is fixed to the crane. [Given:
1 ton = 1000 kg; neglect the weight of the arm.
g = 9.8 m/s2]
Solution:
3. Consider the door shown in the figure, which is seen from an aerial view. The circle on
the left is the hinge (pivot point).
a. Find the Net Torque acting on
the door.
b. Which way will the door open,
up or down?
Solution:
a. τ net=τ 1+ τ 2 +τ 3 + τ 4
τ 1 =r F 1 sinθ=( 0 )( 60 N ) sin 90=0
τ 2=r F 2 sinθ= ( 0.20 m) ( 50 N ) sin 90=10 Nm
τ 3 =r F 3 sinθ =( 0.2 m+0.6 m ) ( 70 N ) sin 90=56 Nm
τ 4=r F 4 sinθ =( 0.2m+0.6 m+0.2 m )( 80 N ) sin 30=40 Nm
Before adding the torque, determine their corresponding direction according to the rotation of the
door.
τ 1 has no rotation since the torque is zero
τ 2 and τ 4: pulling the door upward would make it rotate in the CCW direction (+)
τ 3 : pulling the door downward would make it rotate in the CW direction (-)
τ net=0+10 Nm+ ( −56 Nm ) + 40 Nm=−6 Nm
b. Since the result of the net torque is negative, this means that the door will rotate in
the clockwise or downward direction.
B. STATIC EQUILIBRIUM
16
Static equilibrium occurs when an object is at rest – neither rotating nor translating.
It is analogous to Newton’s 1st Law of motion for rotational system. An object which is not
rotating remains not rotating unless acted on by an external torque. Similarly, an object
rotating at constant angular velocity remains rotating unless acted on by an external torque.
For an object to maintain in static equilibrium, the following conditions must be met:
⃑
1. The net force acting on the object must be zero: ∑ F=0
2. The net torque acting on the object must be zero: ∑ τ=0
This topic will also help you understand important applications in the field of
engineering such as building bridges, the Physics behind crane towers and many more.
Sample Problems:
1. A 0.15kg meterstick is supported at the 50cm mark. A mass of 0.5kg is attached at the
80cm mark.
a. How much mass should be
attached to the 40cm mark to
keep the meterstick horizontal?
b. Determine the supporting force
from the fulcrum on the
meterstick.
Solution:
a. From the 2nd condition of Equilibrium:
∑ τ=0 → τ 1 +τ 2=0
Where τ 1 is the torque caused by the force exerted by mass m
τ 2 is the torque caused by the force exerted by the 0.5kg mass
Hanging mass m would cause the stick to rotate in the CCW direction, thus τ 1 is (+)
Hanging the 0.5kg-mass would cause the stick to rotate in the CW direction, thus τ 2is (-)
∑ τ=τ 1 − τ 2=r 1 F1 sinθ −r 2 F 2 sinθ=0 where F=weigℎt =mg
[(0.10 m)(m)(9.8 m/ s 2) sin 90 ] − [ ( 0.30 m ) (0.5 kg)(9.8 m/ s2 )sin 90 ]=0
(0.98 m 2 /s 2)m− 1.47 Nm=0 2
→ m=1.47 Nm/0.98 m /s ¿=1.5 kg
2
m=1.5 kg
17
Solution:
∑ F y =0
N −W p −W l=0
N=W p +W l=800 N +180 N
N=980 N
F=
( 13 ) ( 5 m )( 800 N ) ( sin 37 ° )+ ( 2.5 m) ( 180 N ) (sin 37 °)
( 0 ) ( 980 Nsinθ ) +
( 5 m ) sin 53°
0+802.42 Nm +270.82 Nm
F=
3.99 m
F=268.98
18 N=f
We have seen how Newton’s Laws of motion is similar to rotational motion. Newton’s
Laws may be stated in terms of rotational motion.
We can derive the equation of Torque in terms of the angular acceleration , α ,from Newton’s
2nd Law of Motion:
⃑ ⃑ multiplying both sides with r
F=m a
⃑ ⃑ ⃑
where r F=τ and a=rα
r F=rm a
τ =( rm ) ( rα )
2 2
τ =m r α ; I =m r
τ =Iα
Rotational Work
To calcula the work done by the torque, we derive it from the translational equation of Work.
W =Fd where d=s=rθ (rotational motion)
W =Frθ; Fr=τ
W =τθ
1
KE= m v 2 v=rω (rotational motion)
2
1 2 1 2 2 2
KE= m(rω) = mr v ; mr =I
2 2
1 2
KE= I ω
2
For vehicles such as cars and bicycles, the tires exert rotational and translational kinetic
energy. Thus, the total kinetic energy is equal to:
1 2 1 2
KE= m v + I ω
2 2
Angular Momentum
L=r x p
Where L is the angular momentum of the object;
r is the distance of the particle from the point of rotation and
p is the linear momentum
Therefore: L=Iω
The higher the angular momentum of the object, the harder it is to stop. Objects with
higher angular momentum have greater orientational stability. That is why in riding a bicycle,
if you are going faster, you will not fall ober easily as when you are going slower.
Conservation of Momentum:
“The momentum of a system will not change unless an external torque is applied.”
Lf =Li (Final momentum=Initial momentum)
Sample Problems:
20
1. Janelle uses a 20cm long wrench to tighten a
nut. The wrench handle is tilted 30º above the
horizontal and Janelle pulls straight down on the
end with a force of 100N. How much torque
does Janelle exert on the nut?
Solution:
τ =r F⊥ =rFsinθ
τ =( 0.20 m )( 100 N ) ( sin 60° ) =17.3 Nm
2. A flywheel of mass 182kg has a radius of 0.62m (assume the flywheel is a hoop).
a. What is the torque required to bring the flywheel from rest to a speed of 120rpm
in an interval of 30 sec?
b. How much work is done in this 30-sec period?
Solution
a. τ =rF=r ( ma )=rm ( rα ) wℎere α =(∆ ω/∆ t )
τ =m r ( )
2 ∆ω
∆t
=mr (
2 ω f − ωi
∆t )
=mr
2 ωf
∆t ( )
wℎere ωi =0(¿ rest )
[ ]
rad
12.57
s
τ =( 182 kg ) ( 0.62 m) =29.31 Nm
30 s
(
ω f = 120 )(
rev 2 πrad
min 1 rev )( 601 minsec )=12.57 rad /sec
b. W =τθ wℎere θ=ωave ∆ t
(
ω +ω
)
W =τ i f ∆ t=( 29.31 Nm )( 12.57 rad / s ¿¿¿ 2 ) ( 30 s )=5,526.4 J
2
3. A 1.20kg disk with a radius of 10.0 cm rolls without slipping. The linear speed of the disk
is 1.41m/s.
a. Find the translational KE.
b. Find the rotational KE.
c. Find the total kinetic energy.
Solution:
1 2 1 2
a. K trans= m v = ( 1.20 kg ) (1.41 m/ s) =1.19 J
2 2
b. K rot =
1
2
I ω
2
=
1 1
(
2 2
m r
2
()
)
v 2 1
r 4
2 1
4
2
= m v = ( 1.20 kg ) (1.41 m/s) =0.596 J
c. K tot =K trans + K rot=1.19 J +0.596 J =1.79 J
Solution:
Since the problem involves the presence of kinetic K , and potential energyU , we use
the conservation of mechanical energy to calculate h.
( )
2 2
1 2.0 m 1 2.0 m
(7.2 kg) + (7.2 kg)( )
2 s ❑ 5 s
ℎf = =0.29 m
9.8 m
(7.2 kg)( 2 )
s
Solution:
2π 2π
a. ω= = =7.85 rad / sec
T 0.8 sec
b. L=Iω=( 1.2 kg m ) (
2 7.85 rad
sec ) 2
=9.42 kg . m / sec
What’s More 22
Direction.Copy the figure in a separate paper and calculate the mass of each item. Show
your solutions.
What I Have Learned
23
Direction. Solve the following problems in a separate paper. Show your solutions
systematically and clearly.
1. A 400.0-N sign hangs from the end of a uniform strut. The strut
is 4.0 m long and weighs 600.0 N. The strut is supported by a
hinge at the wall and by a cable whose other end is tied to the
wall at a point 3.0 m above the left end of the strut. Find the
tension in the supporting cable and the force of the hinge on
the strut.
2. A cylinder of mass m and radius R has a
1 2
moment of inertia of mr . The cylinder
2
is released from rest at a height ℎ on an
inclined plane, and rolls down the plane
without slipping. What is the velocity of
the cylinder when it reaches the bottom of
the incline?
3. A uniform solid cylinder, sphere, and hoop roll without slipping from rest at the top of an
incline. Find out which object would reach the bottom first.
What I Can Do 24
It’s undersTORQUEable!
From the lessons learned, list down 3 sports/events that utilizes the concept of Torque and
briefly explain how this concept is used.
2.
3.
Summary
ROTATIONAL KINEMATICS 25
Angular Displacement θ is the ratio of the arc length (s) to the radius (r) of the circle.
Mathematically, θ=s /r
Angular velocity (ω) is defined as the change in angular displacement (θ) per unit of
∆θ v
time (t). In symbol, ω= =
∆t r
The angular acceleration of a rotating object is the rate at which the angular velocity
changes with respect to time and is given by the equation α =∆ ω/∆ t .
If there is angular acceleration, there will also be tangential acceleration
a tan=rα and 2
a rad =ω r
Analogy between Rotational and Translational Kinematics
ROTATIONAL DYNAMICS
Torque, also called the Moment of Force, is the result of the force that can cause an object
to rotate about an axis. It is a vector quantity. It is the cross product of the vector Force and
the distance from the axis of rotation. Mathematically,τ =rFsinθ
The Torque is dependent on the following factors:
Static equilibrium occurs when an object is at rest – neither rotating nor translating.
For an object to maintain in static equilibrium, the following conditions must be met:
26 ⃑
1. The net force acting on the object must be zero: ∑ F=0
2. The net torque acting on the object must be zero: ∑ τ=0
To calcula the work done by the torque, we derive it from the translational equation of Work
and is equal to W =τθ
1
The Rotational Energy is given by: KE= I ω2
2
For motion involving rotarional and translational kinetic energy, the total energy is equal to
1 2 1
the sum of the two energies. Mathematically, KE= m v + I ω2
2 2
Angular momentum is a quantity that tells us how hard it is to change the rotational motion of
a particular spinning body. L=Iω where I is the moment of inertia of the object.
Conservation of Momentum:
“The momentum of a system will not change unless an external torque is applied.”
Lf =Li (Final momentum=Initial momentum)
Assessment: (Post-Test)
27
Multiple Choice. Answer the question that follows. Choose the best answer from the given
choices.
1. Why are you more stable when riding a bicycle at a faster speed?
A) You have more mass B) The wheels have angular momentum
C) It's not easier to ride at a faster speed D) The bike has more momentum
2. What does the rotational inertia describe?
A) The average position of mass in an extended object.
B) How the mass of an object is distributed
C) How a force can rotate an object.
D) The tendency of an object to move in a straight line.
3. 2600 rev/min is equivalent to which of the following?
A) 2600 rad/s B) 43.3 rad/s C) 273 rad/s D) 60 rad/s
4. A 0.12-m-radius grinding wheel takes 5.5 s to speed up from 2.0 rad/s to 11.0 rad/s.
What is the wheel's average angular acceleration?
A) 9.6 rad/s/s B) 4.8 rad/s/s C) 3.1 rad/s/s D) 1.6 rad/s/s
5. Suppose you are rotating in a chair with 2 equal masses held in each outstretched hand
and you drop them. What happens to your angular velocity?
A) Increase B) Decrease C) Stays the same D) Is lost
6. When seen from below, the blades of a ceiling fan are seen to be revolving
anticlockwise and their speed is decreasing. Select correct statement about the
directions of its angular velocity and angular acceleration.
A) Angular velocity upwards, angular acceleration downwards
B) Angular velocity downwards, angular acceleration upwards
C) Both angular velocity and angular acceleration upwards
D) Both angular velocity and angular acceleration downwards
7. You exert a force on a friend who is holding a 4.0-m-long rope. Now suppose you exert
the same force on your friend, but the friend is holding an 8.0-m-long rope. How will this
affect the rotational acceleration?
A) It will be quartered B) It will be halved C) It will double D) It will quadruple
8. The figure shows scale drawings of four objects, each of the same mass and uniform
thickness, with the mass distributed
uniformly. Which one has the greatest
moment of inertia when rotated about an
axis perpendicular to the plane of the
drawing at point P?
A B C D
9. A person sits on a freely spinning lab stool that has no friction in its axle. When this
person extends her arms,
A) her moment of inertia increases and her angular speed decreases.
B) her moment of inertia decreases and her angular speed increases.
C) her moment of inertia increases and her angular speed increases.
D) her moment of inertia increases and her angular speed remains the same.
10. What is the rotational kinetic energy of the cylinder at t = 2 s?
A)2.0J B)2.5J C)5.0J D) It cannot be determined without knowing the radius
Key to Answers
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