Overview of Oscillations

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Questions and Answers

Which of the following examples represent periodic but not simple harmonic motion?

  • (a) the rotation of earth about its axis. (correct)
  • (d) A freely suspended bar magnet displaced from its N-S direction and released. (correct)
  • (b) motion of an oscillating mercury column in a U-tube.
  • (c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point. (correct)

Which of the following is NOT a characteristic of oscillatory motion?

  • Energy is conserved in this motion.
  • The motion is always uniform. (correct)
  • The restoring force is always proportional and opposite to the displacement.
  • It has a tendency to return to its equilibrium position after being disturbed.

A simple harmonic motion (SHM) is characterized by a restoring force that is:

  • Directly proportional to the displacement and in the same direction.
  • Directly proportional to the displacement and in the opposite direction. (correct)
  • Inversely proportional to the displacement and in the same direction.
  • Inversely proportional to the displacement and in the opposite direction.

What is the relationship between simple harmonic motion and uniform circular motion?

<p>The projection of uniform circular motion onto a diameter is simple harmonic motion. (C)</p> Signup and view all the answers

Which of the following is an example of a periodic motion?

<p>A pendulum swinging back and forth (A)</p> Signup and view all the answers

A bird flapping its wings circles around a clock tower. Which part of the motion is oscillatory?

<p>The bird's motion is periodic while wings flapping is oscillatory. (D)</p> Signup and view all the answers

What is the difference between periodic and oscillatory motion?

<p>Oscillatory motion is always periodic, but periodic motion may or may not be oscillatory. (A)</p> Signup and view all the answers

What is the time period of a periodic motion?

<p>The time it takes for the object to complete one cycle of motion. (B)</p> Signup and view all the answers

What is the SI unit for frequency?

<p>Hertz (B)</p> Signup and view all the answers

Which of the following is an example of oscillatory motion?

<p>A ball bouncing up and down (B)</p> Signup and view all the answers

What does the term 'amplitude' refer to in the context of Simple Harmonic Motion (SHM)?

<p>The maximum displacement of the object from its equilibrium position (A)</p> Signup and view all the answers

What is the relationship between angular velocity (ω) and frequency (f) in SHM?

<p>ω = 2πf (B)</p> Signup and view all the answers

What is the time period (T) of an insect's wing movement if it flaps 144 times in 3 seconds?

<p>0.0208 seconds (D)</p> Signup and view all the answers

Which of the following is NOT an example of periodic motion?

<p>A car accelerating from rest (D)</p> Signup and view all the answers

The motion of a seconds hand on a watch is an example of which type of motion?

<p>Periodic motion (B)</p> Signup and view all the answers

The equation "y = A sin(ωt)" represents the displacement of a particle in SHM. What does the variable 'A' represent?

<p>Amplitude (C)</p> Signup and view all the answers

What is the formula that represents the force acting on the pendulum bob?

<p>F = mg sin θ (D)</p> Signup and view all the answers

How does the time period (T) of oscillation relate to the length of the pendulum (L)?

<p>T ∝ √L (D)</p> Signup and view all the answers

What happens to the center of mass (C.G.) of the hollow sphere as water flows out?

<p>It first goes downward and then rises. (A)</p> Signup and view all the answers

Which factor does NOT influence the time period of the pendulum according to the content?

<p>Mass of the bob (A)</p> Signup and view all the answers

When the hollow sphere is completely filled with water, what is the initial effect on the effective length of the pendulum?

<p>It increases (D)</p> Signup and view all the answers

During oscillation, what describes the change in effective length as water flows out?

<p>It first increases, then decreases. (B)</p> Signup and view all the answers

According to the pendulum's behavior, what determines the oscillation period changes when the water is draining?

<p>The center of mass of the sphere (A)</p> Signup and view all the answers

In the formula T = 2π√(L/g), what does 'g' stand for?

<p>The gravitational acceleration (A)</p> Signup and view all the answers

What is the relationship between spring constant and stability in a car's suspension system?

<p>A higher spring constant increases stability. (B)</p> Signup and view all the answers

What does a greater spring constant imply about a spring?

<p>It requires more force to compress or extend. (D)</p> Signup and view all the answers

How does a softer suspension system affect a car's handling?

<p>It compromises stability and allows more weight transfer. (A)</p> Signup and view all the answers

According to Hooke's law, what does a higher value of k represent in a spring?

<p>A stiffer spring with a greater restoring force. (D)</p> Signup and view all the answers

What is the formula for the resultant amplitude when two waves are superimposed?

<p>D = A^2 + B^2 (B)</p> Signup and view all the answers

What effect does a stiffer suspension have during cornering?

<p>It reduces body roll and weight transfer. (B)</p> Signup and view all the answers

Which of the following represents the phase angle in the context of wave superposition?

<p>tanϕ = B/A (C)</p> Signup and view all the answers

What happens to the height of a car when using a stiffer spring under the same load?

<p>The car will be displaced higher from the ground. (B)</p> Signup and view all the answers

Which factor is most critical for increasing a car's stability through springs?

<p>Utilizing springs with a greater spring constant. (A)</p> Signup and view all the answers

What is the potential energy formula associated with Hooke's law?

<p>PE = 1/2 kx^2 (C)</p> Signup and view all the answers

How does a soft suspension system potentially impact a vehicle's ride quality?

<p>It enhances the ride quality. (C)</p> Signup and view all the answers

In the context of simple harmonic motion, what is the expression for kinetic energy?

<p>KE = 1/2 mv^2 (C)</p> Signup and view all the answers

Which equation describes the relationship between restoring force and displacement in Hooke's law?

<p>F = -kx (B)</p> Signup and view all the answers

What does the variable D represent in the equation z(t) = D sin(ωt + ϕ)?

<p>Resultant amplitude (B)</p> Signup and view all the answers

What is the expression for the phase angle in terms of the sine and cosine components?

<p>ϕ = tan^{-1}(B/A) (D)</p> Signup and view all the answers

What is the natural frequency of a body?

<p>The frequency at which a body vibrates when disturbed from its equilibrium state (A)</p> Signup and view all the answers

Why is it important to understand the natural frequency in engineering?

<p>To build systems that are efficient and less prone to vibrational damage (B)</p> Signup and view all the answers

What occurs during resonance?

<p>Amplitude increases when the driving force frequency is close to the natural frequency (D)</p> Signup and view all the answers

What can best explain the movement of tectonic plates?

<p>Resonance (D)</p> Signup and view all the answers

How does the time period of a swinging pendulum change when a girl standing up swings?

<p>It remains the same since mass does not affect the period (C)</p> Signup and view all the answers

What happens to the frequency of a guitar string as its tension increases?

<p>It increases due to a greater restoring force (A)</p> Signup and view all the answers

What happens to the time period of a swing if it is taken to a higher altitude?

<p>It increases due to reduced gravitational acceleration (D)</p> Signup and view all the answers

Why do aircraft designers ensure that the wings have different natural frequencies from the engine's angular frequency?

<p>To avoid harmonic oscillations that can cause structural damage (B)</p> Signup and view all the answers

Flashcards

Oscillatory Motion

Repeated motion around an equilibrium position, like a pendulum swing.

Periodic Motion

Motion that repeats at regular intervals, like the motion of planets.

Simple Harmonic Motion (SHM)

Periodic motion where the acceleration is proportional to displacement.

Time Period (T)

Time taken for one complete cycle of motion, measured in seconds.

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Frequency (f)

Number of cycles or oscillations per second, measured in Hertz.

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Amplitude (A)

Maximum displacement from the equilibrium position in SHM.

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Angular Velocity (ω)

Rate of change of angular position of a rotating object.

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Displacement in SHM

Position of a particle in SHM at a given time, expressed as y = A sin(ωt).

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Spring constant (k)

A measure of a spring's stiffness; higher k means stiffer spring.

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Resultant Displacement

The combined effect of two waves, expressed as z(t) = D sin(ωt + φ).

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Hooke's Law

Describes the relationship between force and displacement in springs: F = -kx.

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Amplitude

The maximum extent of a periodic wave, given by D = √(A² + B²).

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Restoring force

The force that returns a spring to its equilibrium position when displaced.

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Stiffer spring advantages

Provides greater stability by resisting weight transfer during cornering.

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Phase Angle

The angle that represents the phase difference between two waves, φ = tan⁻¹(B/A).

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Potential Energy (PE)

Energy stored due to displacement from equilibrium, given by PE = (1/2)kx².

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Car stability

The ability of a car to maintain its intended path during driving.

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Softer suspension effects

Allows more body roll and weight transfer, compromising car stability.

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Kinetic Energy (KE)

Energy of motion, generally expressed as KE = (1/2)mv².

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Kinetic Energy (KE)

Energy of a body in motion; for a spring, KE = 1/2 m(−ωA sin(ωt))².

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Hooke's Law

Describes the proportionality of restoring force (F) to displacement (x) in elastic materials: F = -kx.

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Superposition of Waves

The principle where two waves add together, leading to a resultant wave.

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Potential Energy (PE)

Stored energy in a spring; PE = 1/2 kA² cos²(ωt).

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Time Period of Periodic Functions

The minimum duration required for a wave or oscillation to repeat itself.

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Relation between SHM and UCM

Simple Harmonic Motion has a direct link with Uniform Circular Motion.

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Equilibrium Position

The mean position where an object tends to return.

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Energy Conservation in Oscillations

Energy is conserved during oscillatory motion.

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Examples of Periodic but not SHM

Periodic motions that do not follow simple harmonic patterns, like Earth's rotation.

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Force on pendulum (F)

The force acting on a pendulum given by F = mg sin θ.

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Small angle approximation

Assuming θ is very small, sin θ ≈ θ.

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Period of oscillation (T)

The time taken for one complete cycle, T = 2π√(L/g).

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Effective length of pendulum (L)

The length that influences the period of oscillation.

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Center of mass (CM)

The point where the mass of the pendulum is concentrated.

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Water outflow effect

As water flows out, the CM changes and affects T.

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Pendulum dynamics

The behavior of a pendulum based on length and mass.

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Increased then decreased period

As water empties, the period first rises and then falls.

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Natural Frequency

The frequency at which an object vibrates naturally when disturbed.

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Factors Affecting Natural Frequency

Natural frequency depends on size, shape, and material properties of an object.

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Resonance

Increase in amplitude when driving frequency is close to natural frequency.

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Damping Constant (b)

A factor that slows down oscillation; influenced by surrounding fluid.

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Impact of Natural Frequency on Design

Understanding natural frequency helps engineers create safer structures.

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Angular Frequency (ω)

Rate of oscillation, directly related to natural frequency.

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Time Period Change in Pendulum

The period changes with the length of a pendulum and the mass of its bob.

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SHM Parameter Matching

In two SHMs, key parameters like amplitude can remain the same even if frequency differs.

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Study Notes

Overview of Oscillations

  • Oscillation is a repetitive back-and-forth motion around a central point, often measured by time period and frequency.
  • Oscillatory motion is characterized by repetition, and includes periodic and simple harmonic motions.
  • Periodic motion repeats after a fixed time interval, but the path or equilibrium point isn't necessarily the same.
  • Simple harmonic motion (SHM) is a special type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. It's characterized by a sinusoidal (sine or cosine) relationship between displacement and time.

Time Period and Frequency

  • Time period (T) is the time taken for one complete oscillation.
  • Frequency (f) is the number of oscillations per unit time (usually measured in Hertz).
  • The relationship between them is: f = 1/T

Amplitude and Angular Frequency

  • Amplitude (A) is the maximum displacement from the equilibrium position.
  • Angular frequency (ω) is a measure of how quickly an object oscillates and is calculated as ω=2πf or ω = √(k/m), where 'k' is the spring constant and 'm' is the mass.

Characteristics of SHM

  • Restoring force is directly proportional to displacement and directed towards the equilibrium position.
  • Acceleration is directly proportional to displacement but in the opposite direction.
  • The motion is sinusoidal.
  • Total energy is constant, with energy alternating between kinetic energy and potential energy.

Displacement Equation for SHM

  • Displacement (x) as a function of time (t) in SHM can be represented using sine or cosine functions: x = A sin(ωt + φ) or x = A cos(ωt + φ), where φ is the phase constant.

Energy in SHM

  • Total energy remains constant throughout the motion.
  • The energy is exchanged between kinetic and potential energy in a sinusoidal manner.

Types of Oscillation

  • Free oscillations occur without external forces.
  • Forced or driven oscillations are due to external forces with frequencies different from natural frequency.
  • Resonance occurs when the driving frequency is close to the natural frequency, potentially creating large amplitude oscillations.

Simple Pendulum

  • Time period (T) depends only on the length (L) of the pendulum and the acceleration due to gravity (g): T = 2π √(L/g).
  • The period isn't affected by the mass of the bob, only by length and gravity.

Hooke's Law

  • The restoring force of a spring is directly proportional to its displacement: F= −kx, where 'k' is the spring constant.

Examples and Applications

  • Pendulums
  • Springs
  • Musical instruments
  • Bridges
  • Earthquakes and resonance in structures.

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