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General Physics 1

Quarter 2 - Module 3

Periodic Motion

Department of Education ● Republic of the Philippines

Senior High School

General Physics 1 – Grade 12

Alternative Delivery Mode

Quarter 2 – Module 3: Periodic Motion

First Edition, 2020

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Table of Contents

What This Module is About.................................................................................................i

What I Need to Know..........................................................................................................i

How to Learn from this Module..........................................................................................ii

Icons of this Module...........................................................................................................ii

What I Know......................................................................................................................iii

Lesson 1: Periodic Motion

What’s In..................................................................................................................................... 1

What I Need to Know.................................................................................................................. 1

What’s New:................................................................................................................................ 1

What Is It..................................................................................................................................... 2

What’s More: Completing Crossword Puzzle.............................................................................. 3

Lesson 2: Simple Harmonic Motion

What’s In..................................................................................................................................... 4

What I Need to Know.................................................................................................................. 4

What’s New:................................................................................................................................ 4

What Is It..................................................................................................................................... 5

What’s More:............................................................................................................................... 6

What I Have Learned:................................................................................................................. 7

What I Can Do:............................................................................................................................ 7

Lesson 3: Pendulum

What’s In..................................................................................................................................... 8

What I Need to Know.................................................................................................................. 8

What Is It..................................................................................................................................... 8

What’s More:............................................................................................................................... 11

What I Have Learned:................................................................................................................. 11

What I Can Do:............................................................................................................................ 11

Lesson 4: Mechanical Waves

What’s In..................................................................................................................................... 12

What I Need to Know.................................................................................................................. 12

What’s New: Making Waves........................................................................................................ 12

What Is It..................................................................................................................................... 13

What’s More: Solving Sinusoidal Wave Function........................................................................ 17

What I Have Learned: Fill the Waves.......................................................................................... 18

What I Can Do: Geology: Physics of Waves:.............................................................................. 18

Summary......................................................................................................................... 19

Assessment: (Post-Test)................................................................................................. 20

Key to Answers................................................................................................................ 21

References...................................................................................................................... 23

How to Learn from this Module

To achieve the learning competencies 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.

Icons of this Module

i

ii

What I Know

MULTIPLE CHOICES. Directions : Read and understand each item and

choose the letter of the correct answer. Write your answers on a separate sheet of

paper.

1. At which position is the speed of a particle executing SHM greatest?

a. It's extreme position

b. at its equilibrium position

c. at its maximum displacement

d. somewhere between amplitude and equilibrium position

2. At which position is the acceleration of a particle executing SHM equal to zero?

a. at its extreme position

b. at its equilibrium position

c. at its maximum displacement

d. somewhere between its amplitude and equilibrium position

3. The total energy of a simple harmonic oscillator is equal to

k x

2

What does x

represent?

a. any value

b. amplitude of the oscillator

c. equilibrium position of the oscillator

d. position between the maximum displacement and equilibrium position

4. Which type of harmonic motion refers to oscillatory motion with decreasing

amplitude?

a. critically damped

b. overdamped

c. simple

d. under dumped

5. What does a wave carry with it as it travels through a medium?

a. energy b. matter c. water d. wind

6. In which type of wave are the particles of the medium vibrate parallel to the

direction of wave propagation?

a. Longitudinal b. mechanical c. seismic d. transvers

7. Which of the following is an example of a longitudinal wave?

a. gamma ray b. Sound wave c. water wave d. x-ray

8. What is the highest part of the wave called?

a. amplitude b. crest c. trough d. wavelength

9. Which of these is not a characteristic of a wave?

a. amplitude

b. mass

c. period

d. velocity

10. If a wave has a period of 0.25 seconds, what is its frequency?

a.0.25 Hz

b. 1.0 Hz

c. 4.0 Hz.

d. 12 Hz

iii

What Is It

Periodic motion is a motion of an object that regularly repeat—the object

returns to a given position after a fixed time interval. With little thought, we can

identify several types of periodic motion in everyday life. Your car returns to the

driveway each afternoon. You return to the dinner table every night to eat. A bumped

chandelier swing back and forth, returning to the same position at a regular rate. The

Earth return to same position in its orbit around the Sun each year, resulting in the

variation among the four seasons. The Moon return to the same relationship with the

Earth and the Sun, resulting in a full Moon approximately once a month.

Figure 3.1 The Motion of the swing is an example of periodic motion.

A body undergoing periodic motions

always has a stable equilibrium position.

The equilibrium position , otherwise

known as resting position , is the position

assumed by the body when it is not

vibrating. This equilibrium position is

represented by position B of the boy in

the swing and figure 3.1. When the boy is

displaced from its equilibrium position to

position A, a restoring force(gravity) acts

on it to pull it back toward position B.

A restoring force is the force that tends to restore a body from its

displacement to its equilibrium position. By the time the boy reaches position B, the

body has gained kinetic energy, overshoots this position, moves, and stops

somewhere on the other side (Position C). The body is again pulled back towards

equilibrium. Vibrations about this equilibrium position results only from the action of

the restoring force.

The amplitude (A) of vibration is the maximum displacement of a body from

its equilibrium position. This is represented by the displacement from position B to

position A or from position B to position C.

The period (T) of a body in periodic motion is the time required to make a

complete to-and-fro motion. One complete to-and-fro motion is called a cycle.

Referring to figure 3.1, the motion of the swing from position A to position C and

back to position A is one cycle. Period is usually in seconds.

Frequency (f) is the number of the cycle per unit time. It's SI unit is the hertz,

abbreviated as Hz. One (1) hertz equals one cycle per second. Frequency is the

reciprocal of period.

f =

T

Sometimes, angular frequency (ω) is use instead of frequency. Angular

frequency is commonly express in radians per second. The relationship between

angular frequency and frequency is given by

ω = 2 π f ∨ ω =

2 π

T

What’s More

Activity 3.2 Completing Crossword Puzzle

Direction: Complete the crossword puzzle below.

Down

1. It is the number of cycles per

unit of time from its equilibrium

position.

3. It refers to motion that is

repeated at regular intervals of

time.

4. It’s a force that causes body to

accelerate towards the earth.

8. It is the energy possessed by

a body in motion.

Across

2. It is a force that tends to restore a body.
3. It refers to the position assumed by the body when it is not vibrating
4. It refers to one complete to-and-from motion

Equation 3.

Equation 3.

3

1 8

2 4

7

5

6

Equation 3.

a) Do you think this system motion would fall under the classification of simple

harmonic motion? Provide as much evidences as you can for your answer.

b) Do your answers to A and B above matches a spring/mass system or a

pendulum? How so?

c) Do you think this system can be modeled as pendulum, spring, or neither?

What are your reasons for each?

What Is It

A very common type of periodic motion is what we called simple harmonic

motion (SHM). It is a type of periodic motion where the restoring force is

proportional to the displacement of the body from its equilibrium position. This

restoring force act in a direction opposite that of the displacement.

In equation,

F

s

=− kx

where,

F

s

is the restoring force or spring force

x is the displacement from the equilibrium position

k is a proportionality constant

The negative sign simply means that the restoring force and displacement are

oppositely directed. We call this restoring force because it is always directed toward

the equilibrium position and therefore opposite the displacement from equilibrium.

A system that oscillate with SHM is called simple harmonic oscillator. The

simplest form of a simple harmonic oscillator is a body of mass m oscillating on one

end of an elastic spring also known as the mass-spring system.

In the model for simple harmonic motion, consider a block of mass m attached

to the end of a spring, with the block free to move on a horizontal, frictionless surface

Figure 3.2.

Note that the amount of the spring is negligible to the amount of stretching for

compressing force.

An external force can cause object, like spring, to stretch or compressed by a

certain displacement x (figure 3.2). This force is numerically equal to the restoring

force but opposite in direction. Thus,

4

Equation 3.

Figure 3.2. A block attached to a spring moving on a

frictionless surface.

(a) Stretched spring. When the block is displaced to

the right of equilibrium (x > 0), the force exerted by

the spring acts to the left.

(b) When the block is at its equilibrium position (x =

0), the force exerted by the spring is zero.

(c) Compressed spring. When the block is displaced

to the left of equilibrium (x < 0), the force exerted by

the spring acts to the right.

F = kx

The proportionality constant ( k) is what we called the force constant of the

spring. It is the force needed to produce a unit of elongation or compression of the

spring and has the unit of N/m.

The force constant k is measure of the stiffness of the spring. A small value of

k indicates that the spring can be easily stretched or compressed. In other words,

springs with lesser spring constants will have greater displacements than those with

larger spring constants for the same amount of force applied.

Sample Problem 3.

1. An oscillating body takes 0.8 seconds to complete four cycles. What is the

(a)period, (b)frequency, and (c)angular frequency of the body?

Solution:

a. We are asked to determine the time taken to complete one cycle.

T =

time

number of cycles

0.8 s

=0.2 s

b. Substituting the value of T in Equation (3.1),

f =

T

0.2 s

=5. 0 s

− 1

∨5.0 Hz

ω = 2πf = 2(3.14radians)(5. 0 s

− 1

= 31.4 radians /s

Sample Problem 3.

2. A force of 3.0N elongates a spring by 6.0 cm. (a) What is the force constant of

spring? (b) How much force is needed to elongate spring an additional 6.

cm?

Solution:

Using Equation 3.4 and substituting values,

a.

k =

F

x

3.0 N

0.06 m

50 N

m

b.

F = kx =

(

50 N

m

)

( 0.12 m )=6.0 N

What’s More

5

Activity 3.4: Simple Harmonic Motion Problems

Direction: Solve the following problems. Show your complete solutions legibly

and concisely in a separate sheet of paper.

1. It takes 365.25 days for the Earth to complete one revolution around the sun.

Calculate the (a) period, (b) frequency, and (c)angular frequency of Earth as it

What’s In

Lesson

Pendulum

In the previous lesson, we recognized the necessary conditions

for a periodic motion to be classified as a simple harmonic motion. In this lesson, will

learn the concept of simple pendulum and physical pendulum as well as how to

calculate the period and the frequency of simple and physical pendulum.

A pendulum is any object which can swing freely from a pivot point under the

influence of gravity.

What I Need to Know

After this lesson, you should be able to:

1. Calculate the period and the frequency of spring mass, simple pendulum

and physical pendulum (STEM_GP12PM-Ilc-27)

What Is It

The simple pendulum is another mechanical system that exhibit

periodic motion. It consists of a particle-like bob of mass m suspended by a light

string of length L that is fixed at the upper end, as shown in Figure 3.3.

The motion occurs in the vertical plane and it is

driven by

the

gravitational force. We shall show that,

7

25

Figure 3.3 When ϴ is small, a simple pendulum oscillates

in simple harmonic motion about the equilibrium position.

ϴ = 0. The restoring force is –mg sinϴ, the component of

the gravitational force tangent to the arc

provided the angle. is small (less than about 10°), the motion is very close to that of

simple harmonic oscillator.

The forces acting on the bob are the force T exerted by the string and the

gravitational force mg. The tangential component mg sin ϴ. of the gravitational force

always acts toward ϴ=0, opposite the displacement of the bob from the lowest

position. Therefore, the tangential component is a restoring force, and we can apply

Newton’s second law for motion in the tangential direction:

F

t

=− mg sinϴ = m

d

2

s

d t

2

where s is the bob’s position measured along the arc and the negative sign indicates

that the tangential force acts toward the equilibrium (vertical) position.

The period of the motion is T = 2 π

√

L

g

In other words, the period and frequency of a simple pendulum depends only on

the length of the string and the acceleration due to gravity. The simple pendulum can

be used as timekeeper because its period depends only on its length and the local

value of g. It is also a convenient device for making precise measurements of the

freefall acceleration. Such measurements are important because variation in local

values of g can provide information on the location of oil and of other valuable

underground resources.

Based on the equation, the period of a simple pendulum is governed by the

following laws.

1. The period of simple pendulum is directly proportional to square root of its

length.

2. The period is inversely proportional to square root of the acceleration due to

gravity.

3. The period is not dependent of the mass of the bob.
4. The period is independent of the angular amplitude if angular displacement is

small, say less than or equal to 10

0

Sample Problem 3.

A simple pendulum of length 50.0 cm takes 5s to make 10 complete back-

and-forth motion. (a) Find its period. (b) What will be its period when its length is

increased to 200cm?

Solution:

a.

T =

time

number of cycles

5 s

=0.5 s

8

Equation 3.

Manipulating equation 3.5 to solve for I and substituting values,

I =

T

2

mgL

4 π

2

(1.25 ¿¿ 2 )(1.5 kg )(9.8 m / s 2 )(0.5 m )

2

=0.29 kg. m

2

What’s More

Activity 3.

What I Have Learned

Activity 3.8: Self-check Questions

1. What are the things to be considered in describing the motion of a physical

pendulum? The mass of the object, gravity and axis of rotation.

_____________________________________________________________

_____________________________________________________________

2. What are the examples of a physical pendulum? Possible answers:

Chandelier, ceiling fan, lantern

_____________________________________________________________

_____________________________________________________________

3. What are the laws of simple pendulum?

_____________________________________________________________

_____________________________________________________________

What I Can Do

10

Activity 3.7: Problems involving Simple and Physical Pendulum

Direction: Solve the following problems. Show your complete solutions

legibly and concisely in a separate sheet of paper.

1. A simple pendulum is found to vibrate 50 times within 200s. When 1.5m of

its length is reduced to a certain length, it vibrates 50 times in 175s. Find the

original length of the pendulum.

2. A Christmas ball in a shape of a hollow sphere is hung from the tree by a

piece of thread attached to the surface of the ball. The mass and radius of

the ball are 0.105kg and 0.12m, respectively. What will be its period of

oscillation when slightly displaced from its equilibrium position?

(Hint: I =

m r

2

What’s In

Activity 3.9: Create Your Own Problem

Make your own word problem. One problem calculating the period and the

frequency of simple pendulum and one in physical pendulum. Show your complete

solutions to the problems. Be sure that the problems are not taken from the internet

and that they are realistic.

Lesson

Mechanical Waves

We learned from Lesson 1 of this

module that there are many kinds of motion that repeat themselves over and over.

We call this motion as periodic motion or oscillation. As you read through the

concepts of Lesson 1, you realized that periodic motion is used to model a wide

range of physical phenomena. It is also important because it generates waves, which

is the focus of this lesson. Many of the terms and equations we used in Lesson 1 to

2 will be applied in this lesson as we study wave motion especially that of the

mechanical waves.

What I Need to Know

After this lesson, you should be able to:

1. Define mechanical wave, longitudinal wave, transverse wave, periodic wave, and

sinusoidal wave (STEM_GP12PM-IId-31) ;and

2. From a given sinusoidal wave function infer the speed, wavelength, frequency,

period, direction, and wave number (STEM_GP12PMIId-32).

What’s New

Direction: Perform Activity 3.10 and answer the questions. Use a

separate sheet of paper for your answer.

11

Activity 3.10: Making Waves

Learning Target : To generate and describe transverse and longitudinal waves

Materials: string or elastic band, coil or “slinky”

Procedure:

Activity 1: Tape one end of a string to a desk.Then pull the string so it is tight, but lays flat against

the desk. Then generate travelling transverse waves by wiggling the free end of the string up and

down briskly.