3B3 Harmonic Motion Flashcards
Explain laws, attributes, and examples of harmonic motion.
Define:
Simple Harmonic Motion (SHM)
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.
What is the defining characteristic of simple harmonic motion (SHM)?
The restoring force is directly proportional to the displacement and acts in the opposite direction.
This proportionality ensures the system follows Hooke’s law.
Fill in the blank:
Harmonic motion occurs due to a _______ force that tries to return the object to its equilibrium position.
restoring
The restoring force is often proportional to the displacement, as described by Hooke’s law.
True or False:
All oscillatory motion is harmonic motion.
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.
Fill in the blank:
In harmonic motion, the total energy alternates between ______ energy and potential energy.
kinetic
The conservation of energy ensures smooth oscillations between these energy types.
Define:
Damping
Gradual reduction of amplitude in harmonic motion due to energy loss.
Energy loss occurs due to factors like friction or air resistance.
How does harmonic motion differ from chaotic motion?
- 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.
Name two examples of systems that exhibit harmonic motion.
- Pendulums
- Spring-mass systems.
These systems exhibit periodic motion driven by restoring forces.
Define:
Hooke’s Law
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.
What is the spring constant in Hooke’s law?
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).
Fill in the blank:
In Hooke’s law, the restoring force is ______ proportional to the displacement.
directly
The proportionality constant is the spring constant (k).
What is the elastic limit of a material?
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.
What happens if the restoring force exceeds the elastic limit?
The material may undergo permanent deformation or break.
Hooke’s law no longer applies beyond the elastic limit.
Define:
Young’s modulus
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.
Explain how Young’s modulus relates to harmonic motion in a spring-mass system.
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.
How does Hooke’s Law apply to springs?
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.
Define:
Period (T)
The time it takes for one complete oscillation or cycle.
It is measured in seconds (s).
What is the formula for the period of oscillation in SHM using angular frequency?
T = 2π/ω
What is the relationship between angular frequency and linear frequency in SHM?
ω = 2πf
What two main factors determine the frequency of harmonic motion?
- Mass
- Stiffness of the restoring system.
For example, the spring constant in a spring-mass system affects frequency.
Fill in the blank:
Frequency is measured in ______.
Hertz (Hz)
1 Hertz equals 1 cycle per second.
What is the graphical representation of SHM?
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.
Fill in the blank:
The ______ is the number of oscillations completed per second.
frequency
Frequency is the reciprocal of the period (f=1/T).
What is amplitude in harmonic motion?
Maximum displacement from the equilibrium position.
It determines the energy of the oscillation.
What is the formula for Simple Harmonic Motion (SHM)?
x(t) = A sin(ωt + φ)
Amplitude (A) is a constant, it does not depend on time.
True or False:
Increasing the amplitude of harmonic motion affects its frequency.
False
Amplitude affects energy, but the frequency depends on system properties like mass and spring constant.
What happens to the period of oscillation if the mass increases in a spring-mass system?
T_spring = 2π√(m/k)
The period increases because it depends on the square root of the mass.
What is the effect of a stiffer spring (higher k) on the period of oscillation?
The period decreases because the restoring force is stronger.
A higher spring constant reduces the time for each oscillation.
Why is the motion of a vertical spring not affected by gravity?
Gravity shifts the equilibrium position but doesn’t affect the restoring force or frequency.
Frequency depends on k and m, not g.
How does the amplitude of a spring affect its maximum kinetic energy?
Higher amplitude increases the maximum kinetic energy because energy depends on displacement squared.
KE max = (1/2) kA²
How does harmonic motion in a spring relate to energy conservation?
The total energy alternates between potential energy in the spring and kinetic energy of the mass.
Energy remains constant, assuming no damping.
What is a defining characteristic of a pendulum in harmonic motion?
The restoring force is proportional to the sine of the displacement angle.
For small angles ( <15°), the motion approximates SHM.
Fill in the blank:
The frequency of a pendulum depends on its ______ and the acceleration due to gravity.
length
Frequency is independent of mass for a simple pendulum.
Give the formula for the period of a simple pendulum.
Tpendulum = 2π√(L/g)
where L is the length of the pendulum and g is the acceleration due to gravity.
What is the role of damping in pendulum motion?
Damping reduces the amplitude over time, eventually stopping the motion.
Energy is lost to friction and air resistance.
What happens to the frequency of a pendulum if it is taken to the Moon?
The frequency decreases because the gravitational acceleration is lower.
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)
Explain why a pendulum clock may not work correctly at higher altitudes.
Gravitational acceleration decreases, increasing the period of oscillation and slowing the clock.
Pendulum frequency depends on g.
Imagine a child on a swing. What is the amplitude of this harmonic motion?
The maximum height reached by a child on a swing.
The swing’s amplitude represents its furthest displacement from equilibrium.
