SIMPLE HARMONIC MOTION

Subject: 

Physics

Term:

FIRST TERM

Week:

WEEK 9

Class:

SS 2

Topic:

SIMPLE HARMONIC MOTION

Previous lesson: 

The pupils have previous knowledge of

EQUILIBRIUM OF BODIES IN LIQUIDS

that was taught as a topic in the previous lesson

Behavioural objectives:

At the end of the lesson, the learners will be able to

• CONCEPT of S.H.M
• Example of bodies in S.H.M
• Mathematical description of SHM
• Terms used in describing SHM
• Energy conversion in SHM
• Damped oscillation
• Forced vibration in SHM
• Resonance

Instructional Materials:

• Wall charts
• Pictures
• Related Online Video
• Flash Cards

Methods of Teaching:

• Class Discussion
• Group Discussion
• Asking Questions
• Explanation
• Role Modelling
• Role Delegation

REFERENCES:

• new school Physics by MW Anyakoha
• New system physics for senior secondary schools. Dr. Charles Chew.
• Comprehensive Certificate Physics by Olumuyiwa Awe
• Senior School Physics BY PN Okeke, SF Akande
• STAN Physics.

Content:

WEEK 9: DATE ………………………

TOPIC; SIMPLE HARMONIC MOTION

CONTENT

• CONCEPT of S.H.M
• Example of bodies in S.H.M
• Mathematical description of SHM
• Terms used in describing SHM
• Energy conversion in SHM
• Damped oscillation
• Forced vibration in SHM
• Resonance

PERIOD ONE

CONCEPT OF SHM

Simple harmonic motion is an example of periodic motion. A periodic motion is one whose pattern of motion is repeated at regular interval of time.

A body is said to be in Simple Harmonic Motion if it moves along a fixed path such that it acceleration is directly proportional to its displacement from a fixed point and it is also directed toward that fixed point.

Simple harmonic motion can be defined as the motion of a body whose acceleration is always directed towards a fixed point and is proportional to the displacement of the bodies from that point.

Example of bodies in SHM

1. A vibrating simple pendulum
2. A mass at the ended of a vibrating helical spring
3. Oscillation of mercury in a U-tube 4. Motion of the balance wheel of a watch.
4. Motion of prongs of a vibrating tuning fork
5. Motion of a loaded test tube in water

Mathematical definition of SHM

Mathematically, SHM can be defining as;

Where a is the acceleration and y is the displacement.

Introducing a constant,

The displacement of bodies in SHM simulate the sinusoidal change describe by the sine

curve.

y

Terms for describing a SHM

1. Amplitude (A). this the maximum displacement from the equilibrium position.

A

1. Period (T); this is the time taken for the body in SHM to complete one oscillation. If a body in SHM complete n cycles/oscillations in time t, the period of the SHM is given as.

The S.I unit of period is seconds

1. Frequency (f); this is the number of cycles completed by a body in SHM in one seconds. The S.I unit of frequency is Hertz (s^-1). Frequency can also be defined as the reciprocal of period,
2. Angular frequency; this is the ratio of one complete cycle to the period of the SHM.

Angular frequency is sometime referred to as angular speed. It S.I unit is rad/s. another unit for is rev/min

1. Displacement (y); the displacement of a body in SHM simulate the sine curve and it is given as;

ϴ

is the angular displacement and it is given as t)

Where y is the displacement, A is the amplitude, is the angular frequency and t is time.

EVALUATION

1. State five example of bodies in SHM
2. Define the following terms (i) amplitude (ii) frequency (iii) period

PERIOD TWO.

SPEED AND ACCELERATION OF BODIES IN SHM

1. Speed (v); this is the rate of change of displacement/distance. Since this is not a uniform motion, we result to differential calculus Speed of a body in SHM can also be given as,
2. body in SHM will have it maximum value when sint =1

7. Acceleration (a); this is the rate of change of velocity.

(students who had not taken lesson in differential calculus should not bother about the derivation. But they should take note of the result)

EXPERIMENT

• • To determine acceleration due to gravity g using simple pendulum.
  • To determine the force constant of a helical spring

CLASSWORK

• 1. A body executing simple harmonic motion has an angular velocity of 22rads^-1. If it has a maximum displacement of 10cm. what is its maximum linear velocity? 1m

Simple pendulum

This consists of a small mass attached to the end of a string.

O

C

B

A

1. is the equilibrium position. As the body passes through this position it kinetic energy is maximum.

At B, speed is maximum kinetic energy is maximum potential energy is zero

At A and C, the bodyis temporarily at rest. Speed is zero as well as kinetic energy. But the potential energy at this point is maximum.

The period of oscillation of a simple pendulum

• 1. Period is directly proportional to the square root of the length of the string Period increases with length.
  2. Is independent on the mass of the bob
  3. Is inversely proportional to the square root of the acceleration due to gravity.

A body at the end of a vibrating helical spring.

This is another example of a simple a harmonic motion. It consists of a mass attached to the end of a spring.

The period of the mass vibrating at the end of the helical spring is

1. Directly proportional to the square root of the mass Period increases with mass
2. Inversely proportional to the square root of the force constant of the spring. The period of a vibrating mass at the end of a helical spring is given as EVALUATION.
3. The number of cycle per seconds completed by a body in SHM is called —- (a) period

(b) amplitude (c) angular frequency (d) frequency

1. Which of these is not true about bodies in SHM? (a) acceleration is proportional to displacement (b) acceleration act in opposite direction to the displacement (c) acceleration is directed toward the fixed point
2. A simple pendulum has a period of 4.2 s. when the length is shortened by 1m, the period is 3.7 s. calculate the original length of the string. 9a) 74.5 m (b) 3.2 m (c) 2.7

m (d) 1.8 m (ACEDEX, 2011)

Period 3: ENERGY IN SHM

A body in simple harmonic motion undergoes displacement as a result of a restoring force acting on its toward the equilibrium position. Energy is always involved when a body moves through a distance under the action of a force.

Recall work done = force x distance

Consider a vibrating mass at the end of a helical spring; if an average force of ½F act on the mass to cause a displacement of y

Work done = energy = average force x distance

E = ½ F x y

This is equivalent to the elastic potential energy stored in the spring.

But F = Ky

Where k – force constant or elasticity constant y – extension/ displacement

Substituting this into (vi)

The potential energy of the mass is maximum when y = amplitude

… … … … … (viii)

The kinetic energy of the mass is given as

But from eqn (vb)

Therefore, KE at displacement y is given as

Maximum kinetic energy will occur at the equilibrium position (i.e when y = 0)

EVALAUTION

1. What is the angular speed of a body vibrating at 50cycles per second? (a) 200∏rads^-1

(b) 100∏ rads^-1 (b) 50 rads^-1 (d) 0.01 rads^-1

1. If a body moving with SHM has an angular velocity of 50rad/s and amplitude 10cm, calculate it linear velocity.
2. A body in SHM has an amplitude of 10 cm and a frequency of 100Hz calculate (i) acceleration at maximum displacement (b) period of oscillation (c) velocity at the centre of the motion

PERIOD 4

Simple pendulum

O

C

B

A

The energy of the bob at B is entirely kinetic (equilibrium position)

The energy of the bob at A and C is entirely potential (the bob is temporarily at rest at these points).

At any point between A and B or C and B, the energy is the sum of the potential and the kinetic energy of the bob at that position.

Damped oscillation

SHM is an hypothetical motion in which energy has been taking to be constant through the motion and the amplitude does not change. However, a real-life situation is the damp oscillation in which amplitude die out with time due to air resistance. Energy of the system also depreciate with time.

In damped harmonic oscillation, the amplitude decreases with time until it is zero.

To maintain an oscillation that would have been damped in simple harmonic motion,an external periodic force is applied. This is called forced vibration

Forced vibration is a vibration resulting from the action of an external periodic force on an oscillating body.

Resonance; this is a phenomenon in which the frequency of the external oscillator coincides with the natural frequency of a body thereby making the body to vibrate with a large amplitude.

Resonance explains why sometime at a radio playing some tunes could make a tumbler on the same table to shake visibly.

EVALUATION

1. A simple pendulum has a period of 3.0 s. If the value of g =9.9 m/s². Calculate the length of the pendulum.
2. An object moving with SHM has amplitude 5 cm and frequency 50Hz. Calculate (i) period of the oscillation (ii) acceleration at the middle and end of the oscillation (iii) velocity at the middle and at the end of the oscillation
3. Define the following (I) damped oscillation (ii) forced oscillation (iii) resonance

WEEK END ASSIGNMENT

1. Sketch the curve for displacement speed and acceleration and state the phase difference between them.
2. Beginning from V = Ashow that V
3. Describe an experiment to verify the variation of the period of a simple pendulum with length of the pendulum.
4. The period of a simple pendulum is 3.45 s. when the length of the pendulum is shortened by 1 m, the period is 2.81 s. calculate (a) the original length of the pendulum (b) the acceleration due to gravity

READING ASSIGNMENT

Students should read up Simple Harmonic Motion in their text books.