Physics Chapter on Pendulums and SHM

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Questions and Answers

If Miss Dema stands up while swinging, what happens to the period of the swing?

  • The period becomes unpredictable.
  • The period decreases. (correct)
  • The period stays the same.
  • The period increases.

What happens to the period of the swing if Dema swings at a higher altitude?

  • The period stays the same. (correct)
  • The period increases.
  • The period decreases.
  • The period becomes unpredictable.

If Dema's friend joins her on the swing, what happens to the period?

  • The period decreases.
  • It depends on the friend's mass.
  • The period increases. (correct)
  • The period stays the same.

What is the formula for the time period (T) of a simple pendulum?

<p>$T = 2\pi\sqrt{\frac{l}{g}}$ (D)</p> Signup and view all the answers

What is the relationship between the time period of a simple pendulum and the acceleration due to gravity?

<p>They are inversely proportional. (C)</p> Signup and view all the answers

According to Hooke's law, what is the relationship between the force (F) exerted by a spring and the displacement (x) from its equilibrium position?

<p>F is directly proportional to x. (C)</p> Signup and view all the answers

What would happen to the time period of a simple pendulum if the mass of the bob were doubled?

<p>The period would remain the same. (B)</p> Signup and view all the answers

Calculate the spring constant (k) of a spring if a force of 0.4 N is required to stretch it by 0.1 m.

<p>4 N/m (A)</p> Signup and view all the answers

What is the amplitude of the resultant SHM described by the displacement equation: x = 6cos𝜔t + 8sin𝜔t in meters?

<p>10 (D)</p> Signup and view all the answers

What is the phase angle (in radians) of the resultant SHM described by the displacement equation: x = 6cos𝜔t + 8sin𝜔t?

<p>0.93 (D)</p> Signup and view all the answers

What is the equation for the resultant displacement z(t) of two waves, x(t) and y(t), given by x(t) = A sin t and y(t) = B cos t?

<p>z(t) = D sin(t +  ) (B)</p> Signup and view all the answers

What is the formula for calculating the resultant amplitude D of the combined waves x(t) and y(t)?

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

What is the relationship between the restoring force F and the displacement x in a simple harmonic oscillator, according to Hooke's Law?

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

What is the formula for potential energy (PE) in a simple harmonic oscillator, expressed in terms of the spring constant k and displacement x?

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

Given the velocity of a simple harmonic oscillator is v = -ωA sin ωt, what is the formula for the kinetic energy (KE) of the oscillator?

<p>KE = 1/2m(ωA sin ωt)^2 (B)</p> Signup and view all the answers

What is the time period of a combination of periodic functions?

<p>The minimum time period among the combined functions (D)</p> Signup and view all the answers

What is the time period (T) of a motion that completes one full oscillation?

<p>The time taken for one complete oscillation (C)</p> Signup and view all the answers

Which of the following best describes the frequency (f) of a periodic motion?

<p>The number of oscillations completed in one second (C)</p> Signup and view all the answers

What is the unit of frequency in the context of periodic motion?

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

Which of the following is an example of periodic motion that is not oscillatory?

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

In simple harmonic motion (SHM), how is the displacement of a particle represented?

<p>y = A sin(ωt) (B)</p> Signup and view all the answers

What is the amplitude (A) in the context of SHM?

<p>The maximum displacement from the mean position (D)</p> Signup and view all the answers

If a pendulum has a frequency of 2 Hertz, what is its period (T)?

<p>0.5 seconds (D)</p> Signup and view all the answers

In the formula $ heta = 2 heta_{max}$ for angular displacement, what does $ heta_{max}$ represent?

<p>The maximum angular displacement (D)</p> Signup and view all the answers

What is the maximum displacement of a particle in SHM when it reaches extreme position A?

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

What is the formula for maximum velocity in SHM?

<p>v_max = ω A (B)</p> Signup and view all the answers

What is the equation for acceleration in SHM?

<p>a = -ω^2 A sin ωt (A)</p> Signup and view all the answers

What is the maximum acceleration at an extreme position if the amplitude is 10 cm and maximum speed is 5 m/s?

<p>250 m/s² (D)</p> Signup and view all the answers

If a particle is at maximum displacement in the positive x-direction, what is the phase φ?

<p>0 (A)</p> Signup and view all the answers

As the particle crosses its equilibrium position at maximum velocity, which phase constant φ applies?

<p>π/2 (C)</p> Signup and view all the answers

What is the formula representation of displacement in SHM at any time t?

<p>x = A cos ωt (A)</p> Signup and view all the answers

What describes the direction of motion and position of a particle in SHM at a given time?

<p>Phase (A)</p> Signup and view all the answers

What is the expression that represents the acceleration of a particle in simple harmonic motion related to its displacement?

<p>β = -ω²y (C)</p> Signup and view all the answers

If the displacement of a particle in SHM is given by y = 0.2 sin(50πt + 1.57), which of the following represents the amplitude of this motion?

<p>0.2 m (C)</p> Signup and view all the answers

A particle in SHM has a maximum displacement of 4 cm. If its acceleration is 3 cm/s² at a distance of 1 cm from the mean position, what will be its velocity at 2 cm from the mean position?

<p>6 cm/s (B)</p> Signup and view all the answers

For the SHM equation x = 0.25 cos(π/8 t), what is the angular frequency?

<p>π/8 rad/s (B)</p> Signup and view all the answers

In the equation V = ωA√(1 - (x²/A²)), which variable represents the displacement from the mean position?

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

What is the maximum acceleration of a particle in SHM if its maximum speed is known to be 10 cm/s and its amplitude is 5 cm?

<p>50 cm/s² (A)</p> Signup and view all the answers

Which relationship between acceleration (β) and displacement (y) is incorrect for a particle undergoing simple harmonic motion?

<p>β = 3y (D)</p> Signup and view all the answers

What will be the period of oscillation for a spring-mass system given its angular frequency ω = 4 rad/s?

<p>π/4 s (D)</p> Signup and view all the answers

What happens to the period of a simple pendulum if a little mercury is drained off?

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

What type of oscillation occurs when a body vibrates with its own natural frequency?

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

If the total energy of a system executing simple harmonic motion is 5.83 J, what does this imply about the maximum speed of the object?

<p>It is dependent on the spring constant. (B)</p> Signup and view all the answers

For a block weighing 4 kg that extends a spring by 0.16 m, what will happen if a lighter body is hung from the same spring?

<p>The period of vibration will increase. (C)</p> Signup and view all the answers

Which angular frequency is defined as the frequency at which a body oscillates if suddenly disturbed?

<p>Natural frequency (ω) (C)</p> Signup and view all the answers

What is an example of forced or driven oscillation?

<p>A person pushing a child on a swing (B)</p> Signup and view all the answers

What is the relationship between amplitude and total energy in a simple harmonic oscillator?

<p>Total energy is directly proportional to the square of the amplitude. (A)</p> Signup and view all the answers

What would be the effect of a second force with a frequency different from the natural frequency on a body?

<p>It will cause forced oscillation in the body. (C)</p> Signup and view all the answers

Flashcards

Periodic Motion

Motion that repeats at regular intervals over time.

Oscillatory Motion

Periodic motion that moves back and forth around a central point.

Simple Harmonic Motion (SHM)

A type of oscillatory motion where the motion can be described as a sine wave.

Time Period (T)

The time taken for one complete oscillation or rotation.

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Frequency (f)

The number of oscillations that occur in one second, measured in hertz.

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Amplitude (A)

Maximum displacement from the mean or equilibrium position in motion.

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Angular Velocity (ω)

The rate of change of angular displacement, measured in radians per second.

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Displacement Equation of SHM

y = A sin(ωt) defines position of SHM at any time.

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Maximum Displacement

The furthest distance from equilibrium position in SHM, θ=90°.

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Velocity in SHM

The speed of a particle in SHM, given by v = ωA cos(ωt).

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Acceleration in SHM

The change in velocity of a particle, a = -ω²A sin(ωt).

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Maximum Velocity

The highest speed reached in SHM, V max = ωA.

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Maximum Acceleration

The highest acceleration at extreme positions, a max = ω²A.

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Phase Constant (φ)

The value that shows the initial position and direction in SHM.

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Phase of Motion

Describes a particle's position and direction at a specific time, φ + ωt.

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Sine and Cosine Representation

Displacement, velocity, and acceleration graphs represent SHM behavior.

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Resultant Displacement

The overall displacement when two waves combine, expressed as z(t) = D sin(ωt + φ).

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Phase Difference

The angle showing how much one wave leads or lags behind another, found using tanφ = B/A.

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Resultant Amplitude (D)

The amplitude of the combined wave, calculated as D = √(A² + B²).

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Potential Energy (PE) in SHM

The energy stored due to the position in SHM, given by PE = 1/2 kx².

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Kinetic Energy (KE) in SHM

The energy of motion in SHM, represented as KE = 1/2 mv².

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Displacement of SHM

Describes the position of a particle over time, e.g., x = 6 cos(ωt) + 8 sin(ωt).

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Spring Constant (k)

A measure of a spring’s stiffness in Hooke’s Law, indicating force per unit displacement.

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Restoring Force

The force that brings a displaced object back towards its equilibrium position, given by F = -kx.

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Effect of Height on Time Period

Swinging at a higher altitude increases the time period due to changes in gravitational force.

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Effect of Mass on Swing

Adding mass does not affect the time period of a simple pendulum.

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Hooke's Law

States that the force exerted by a spring is directly proportional to its displacement.

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Newton's Second Law

States that the force is the product of mass and acceleration (F=ma).

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Time Period of a Pendulum

T is determined by the length of the pendulum and gravitational acceleration (l and g).

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Effect of Gravity on Time Period

Time period increases when taken to the Moon due to lower gravitational pull.

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Spring Constant Calculation

The spring constant can be calculated using F = kx, where F is force and x is displacement.

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Potential Energy in SHM

In SHM, potential energy changes with the position of the particle.

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Total Energy (E)

Energy stored in a spring-mass system during motion.

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Maximum Velocity (Vmax)

Highest speed attained by an object in SHM.

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Period (T)

Time taken for one complete cycle of motion.

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Angular Frequency (ω)

Rate at which an object oscillates, in radians per second.

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Maximum Acceleration (Amax)

Highest acceleration of an object in SHM.

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Free Oscillation

Natural motion of a system without external force.

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Displacement (x)

The distance of the particle from its mean position in SHM.

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Velocity (V) in SHM

The speed of a particle at any point in SHM, given by V = ωA cos(ωt).

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Acceleration (β) in SHM

The acceleration of the particle, expressed as β = -ω²y, where y is displacement.

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Study Notes

Summary of Oscillatory Motion

  • Oscillatory motion is a motion in which a body moves back and forth repeatedly about a fixed point in a definite time interval.
  • Periodic motion is any motion that repeats itself at regular intervals; it can be oscillatory but not necessarily so.
  • Oscillatory motion is always periodic.

Characteristics of Oscillatory Motion

  • Tendency to return to equilibrium or mean position after disturbance.
  • Restoring force is always proportional and opposite to displacement.
  • Energy is conserved in this motion.

Simple Harmonic Motion (SHM)

  • SHM is a special type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and directed towards it.
  • SHM can be represented as the projection of uniform circular motion onto a diameter of the circle.
  • Displacement equation: x = A sin(ωt + φ) or x = A cos(ωt + φ)
    • A = amplitude
    • ω = angular frequency
    • t = time
    • φ = phase constant

Time Period and Frequency

  • Time period (T): the time taken for one complete oscillation.
  • Frequency (f): the number of oscillations completed in one second.
    • f = 1/T

Energy in SHM

  • Total energy (TE) is the sum of potential energy (PE) and kinetic energy (KE).
  • At mean position, KE is maximum and PE is minimum.
  • At extreme position, PE is maximum and KE is zero.
  • Total energy remains constant throughout the motion.

Factors affecting Time Period

  • Length of the pendulum: Time period is directly proportional to the square root of the length.
  • Acceleration due to gravity: Time period is inversely proportional to the square root of the acceleration due to gravity.
  • Mass of the oscillating object: Time period is independent of the mass of the object.

Resonance

  • Resonance is a phenomenon in which the amplitude of oscillation increases when the frequency of the driving force is close to the natural frequency of the oscillator.
  • For maximum amplitude displacement:
    • ω = ω₀ (natural frequency)
    • damping constant (b) should be minimum.

Examples of Oscillatory Motion

  • Pendulum
  • Spring-mass system
  • Swinging
  • Motion of a simple pendulum
  • Motion of an oscillating mercury column in a U-tube
  • Motion of a ball bearing inside a smooth curved bowl
  • Motion of hands of a clock
  • The oscillation of a swing in a park
  • The vibration of strings in a guitar

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