3B3 Harmonic Motion

Question 1

Define:

Simple Harmonic Motion (SHM)

Answer 1

Repetitive movement in which an object oscillates back and forth around an equilibrium position, typically under the influence of a restoring force.

Examples include the swinging of a pendulum and the vibration of a spring.

Question 2

What is the defining characteristic of simple harmonic motion (SHM)?

Answer 2

The restoring force is directly proportional to the displacement and acts in the opposite direction.

This proportionality ensures the system follows Hooke’s law.

Question 3

Fill in the blank:

Harmonic motion occurs due to a _______ force that tries to return the object to its equilibrium position.

Answer 3

restoring

The restoring force is often proportional to the displacement, as described by Hooke’s law.

Question 4

True or False:

All oscillatory motion is harmonic motion.

Answer 4

False

Oscillatory motion may not always follow the principles of harmonic motion, such as being proportional to displacement. For example, planets moving around the sun show oscillatory motion but not harmonic motion.

Question 5

Fill in the blank:

In harmonic motion, the total energy alternates between ______ energy and potential energy.

Answer 5

kinetic

The conservation of energy ensures smooth oscillations between these energy types.

Question 6

Define:

Damping

Answer 6

Gradual reduction of amplitude in harmonic motion due to energy loss.

Energy loss occurs due to factors like friction or air resistance.

Question 7

How does harmonic motion differ from chaotic motion?

Answer 7

• Harmonic motion is predictable and follows regular oscillations.
• Chaotic motion is unpredictable and lacks a fixed pattern.

Harmonic motion is governed by linear restoring forces, unlike chaotic systems.

Question 8

Name two examples of systems that exhibit harmonic motion.

Answer 8

1. Pendulums
2. Spring-mass systems.

These systems exhibit periodic motion driven by restoring forces.

Question 9

Define:

Hooke’s Law

Answer 9

The restoring force is directly proportional to the displacement

The equation for Hooke’s law is F=−kx. The negative sign indicates the force acts opposite to displacement.

Question 10

What is the spring constant in Hooke’s law?

Answer 10

The spring constant (k) is a measure of the stiffness of a spring or elastic material.

It has units of Newtons per meter (N/m).

Question 11

Fill in the blank:

In Hooke’s law, the restoring force is ______ proportional to the displacement.

Answer 11

directly

The proportionality constant is the spring constant (k).

Question 12

What is the elastic limit of a material?

Answer 12

Maximum stress a material can withstand without undergoing permanent deformation.

Beyond this point, the material will not return to its original shape when the stress is removed.

Question 13

What happens if the restoring force exceeds the elastic limit?

Answer 13

The material may undergo permanent deformation or break.

Hooke’s law no longer applies beyond the elastic limit.

Question 14

Define:

Young’s modulus

Answer 14

Measure of a material’s stiffness, defined as the ratio of stress to strain in the elastic deformation region

It is measured in Pascals (Pa) and indicates how much a material resists deformation under tension or compression.

Question 15

Explain how Young’s modulus relates to harmonic motion in a spring-mass system.

Answer 15

It determines the stiffness of a material, which affects the spring constant (k) in Hooke’s law.

The spring constant for a material can be expressed as:k=(EA)/L , where A is the cross-sectional area, and L is the length. A stiffer material results in a higher spring constant, increasing the system’s frequency.

Question 16

How does Hooke’s Law apply to springs?

Answer 16

The stretching of a spring when a weight is hung on it obeys the properties of Hooke’s Law.

The spring exerts a restoring force proportional to its displacement.

Question 17

Define:

Period (T)

Answer 17

The time it takes for one complete oscillation or cycle.

It is measured in seconds (s).

Question 18

What is the formula for the period of oscillation in SHM using angular frequency?

Answer 18

T = 2π/ω

Question 19

What is the relationship between angular frequency and linear frequency in SHM?

Answer 19

ω = 2πf

Question 20

What two main factors determine the frequency of harmonic motion?

Answer 20

1. Mass
2. Stiffness of the restoring system.

For example, the spring constant in a spring-mass system affects frequency.

Question 21

Fill in the blank:

Frequency is measured in ______.

Answer 21

Hertz (Hz)

1 Hertz equals 1 cycle per second.

Question 22

What is the graphical representation of SHM?

Answer 22

A sine or cosine wave, showing displacement, velocity, or acceleration as a function of time.

In SHM, displacement follows the equation x(t)=Asin(ωt+ϕ), where A is the amplitude, ω the angular frequency, and ϕ the phase constant.

Question 23

Fill in the blank:

The ______ is the number of oscillations completed per second.

Answer 23

frequency

Frequency is the reciprocal of the period (f=1/T).

Question 24

What is amplitude in harmonic motion?

Answer 24

Maximum displacement from the equilibrium position.

It determines the energy of the oscillation.

Question 25

What is the **formula for Simple Harmonic Motion** (SHM)?

Answer 25

x(t) = A sin(ωt + φ) ## Footnote Amplitude (A) is a constant, it does not depend on time.

Question 26

# True or False: Increasing the **amplitude of harmonic** motion affects its **frequency**.

Answer 26

False ## Footnote Amplitude affects energy, but the frequency depends on system properties like mass and spring constant.

Question 27

What happens to the **period** of oscillation if the **mass** increases in a **spring-mass system**?

Answer 27

T_spring = 2π√(m/k) ## Footnote The period increases because it depends on the square root of the mass.

Question 28

What is the effect of a **stiffer spring (higher k)** on the **period of oscillation**?

Answer 28

The **period decreases** because the restoring force is stronger. ## Footnote A higher spring constant reduces the time for each oscillation.

Question 29

Why is the motion of a **vertical spring** not affected by gravity?

Answer 29

Gravity **shifts the equilibrium position** but doesn’t affect the restoring force or frequency. ## Footnote Frequency depends on k and m, not g.

Question 30

How does the amplitude of a spring affect its **maximum kinetic energy**?

Answer 30

**Higher amplitude increases the maximum kinetic energy** because energy depends on displacement squared. ## Footnote KE max = (1/2) kA²

Question 31

How does **harmonic motion** in a **spring** relate to energy conservation?

Answer 31

The total energy alternates between **potential energy** in the spring and **kinetic energy** of the mass. ## Footnote Energy remains constant, assuming no damping.

Question 32

What is a defining characteristic of a **pendulum** in **harmonic motion**?

Answer 32

The restoring force is **proportional to the sine of the displacement angle**. ## Footnote For small angles ( <15°), the motion approximates SHM.

Question 33

# Fill in the blank: The **frequency of a pendulum** depends on its \_\_\_\_\_\_ and the acceleration due to gravity.

Answer 33

length ## Footnote Frequency is independent of mass for a simple pendulum.

Question 34

Give the formula for the **period** of a **simple pendulum**.

Answer 34

T_pendulum = 2π√(L/g) ## Footnote where L is the length of the pendulum and g is the acceleration due to gravity.

Question 35

What is the role of **damping in pendulum** motion?

Answer 35

Damping **reduces the amplitude over time**, eventually stopping the motion. ## Footnote Energy is lost to friction and air resistance.

Question 36

What happens to the **frequency** of a **pendulum** if it is taken to the Moon?

Answer 36

The frequency **decreases** because the gravitational acceleration is lower. ## Footnote The frequency of a simple pendulum depends on the acceleration due to gravity (g) and the length of the pendulum (L) as given by f= (1/2π)√(g/L)

Question 37

Explain why a **pendulum clock** may not work correctly at higher altitudes.

Answer 37

**Gravitational acceleration decreases**, increasing the period of oscillation and slowing the clock. ## Footnote Pendulum frequency depends on g.

Question 38

Imagine a child on a swing. What is the **amplitude** of this harmonic motion?

Answer 38

The **maximum height** reached by a child on a swing. ## Footnote The swing’s amplitude represents its furthest displacement from equilibrium.