Oscillatory and Simple Harmonic Motion

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Questions and Answers

What is the maximum acceleration calculated using the given parameters?

  • 0.75 m/s²
  • 0.5 m/s²
  • 2.00 m/s²
  • 1.25 m/s² (correct)

Which equation is used to express the position as a function of time?

  • x = A tan(vt + f)
  • x = A cos(vt + f) (correct)
  • x = A sin(vt + f)
  • x = A e^(vt + f)

What would be the form of the velocity function if the initial velocity is vi = -0.100 m/s?

  • v(t) = -0.250 sin(5.00t) (correct)
  • v(t) = 0.300 sin(5.00t)
  • v(t) = 0.250 sin(5.00t)
  • v(t) = -0.300 sin(5.00t)

Which part of the solution remains unchanged regardless of the initial conditions?

<p>The period of oscillation (B)</p> Signup and view all the answers

What is the expression for the acceleration as a function of time?

<p>a(t) = -1.25 cos(5.00t) (C)</p> Signup and view all the answers

What does the red drive belt transfer to the sewing machine mechanism?

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

What type of motion results from the oscillation of the treadle?

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

How does the movement of one's feet on the treadle affect the sewing machine?

<p>Causes circular motion of the drive wheel (A)</p> Signup and view all the answers

What type of oscillation does the shadow of the ball cast on the screen exhibit?

<p>Simple harmonic motion (A)</p> Signup and view all the answers

In the experimental arrangement, what does the ball's movement on the turntable represent?

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

What angle does the line OP make with the x-axis at time t=0 in the reference circle?

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

What is the radius of the circle in the context of the ball and turntable?

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

What relationship does the experimental arrangement illustrate?

<p>Between simple harmonic and uniform circular motion (A)</p> Signup and view all the answers

What does the variable 'K' represent in the context of the block-spring system?

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

At which point in simple harmonic motion is the kinetic energy at its maximum?

<p>At equilibrium position (A)</p> Signup and view all the answers

Which of the following combinations correctly identifies potential energy at maximum displacement?

<p>Maximum potential energy and minimum kinetic energy (B)</p> Signup and view all the answers

What does the variable 'v' symbolize in the equations for the block-spring system?

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

How is the total mechanical energy in a block-spring system expressed?

<p>Kinetic energy plus potential energy (D)</p> Signup and view all the answers

What is the phase constant calculated from dividing the two equations?

<p>0.121π (A)</p> Signup and view all the answers

What does the variable 'S' in the provided figures likely represent?

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

How is the new maximum speed of the oscillator calculated?

<p>By using $v_{max} = vA$ (B)</p> Signup and view all the answers

Which energy is at its lowest value when the spring is compressed or stretched to its maximum displacement?

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

What is the expression for maximum acceleration derived from the equations?

<p>a_max = 1.35 m/s² (B)</p> Signup and view all the answers

What happens to the potential energy as the block in the spring system moves towards its equilibrium position?

<p>It decreases steadily (B)</p> Signup and view all the answers

What is the role of the energy approach compared to motion variables in solving problems?

<p>Energy approach is typically simpler. (A)</p> Signup and view all the answers

In the block-spring system, what is implied by the variable 't'?

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

How is the frequency of vibration of the car determined after hitting a pothole?

<p>Through the spring force constant and total mass of occupants (D)</p> Signup and view all the answers

During which phase of motion is the spring potential energy at maximum?

<p>At maximum compression (C)</p> Signup and view all the answers

What is the expression for the position of the oscillating system in SI units?

<p>x = 0.0539 cos(5.00t + 0.121π) (D)</p> Signup and view all the answers

What factor is NOT included in the force constant of the car's springs?

<p>Number of springs (A)</p> Signup and view all the answers

What is the mathematical representation of the velocity of the oscillator?

<p>v = 0.269 sin(5.00t + 0.121π) (C)</p> Signup and view all the answers

What is the relationship between the torque exerted by the twisted wire and the angular position of the object?

<p>The torque is directly proportional to the angular position. (C)</p> Signup and view all the answers

What does the symbol 'k' represent in the context of a torsional pendulum?

<p>The torsion constant of the support wire. (B)</p> Signup and view all the answers

What effect do nonconservative forces have on oscillatory motion?

<p>They cause the mechanical energy of the system to diminish over time. (A)</p> Signup and view all the answers

What is the formula for the period of a torsional pendulum?

<p>T = 2π√(k/I) (A)</p> Signup and view all the answers

In a torsional pendulum, what happens if the elastic limit of the wire is exceeded?

<p>The wire becomes permanently twisted and cannot oscillate. (C)</p> Signup and view all the answers

What is the effect of damping on oscillatory systems?

<p>The amplitude of the oscillations decreases over time. (B)</p> Signup and view all the answers

How can the torsion constant 'k' be determined experimentally?

<p>By applying a known torque and measuring the angle of twist. (A)</p> Signup and view all the answers

Which of the following statements is true regarding the motion of a torsional pendulum?

<p>It behaves as a simple harmonic oscillator without restrictions. (A)</p> Signup and view all the answers

What is the direction of the acceleration of a particle moving in a circle of radius A?

<p>Inward toward the center (C)</p> Signup and view all the answers

What is the magnitude of the acceleration of a particle moving in a circle of radius A with angular speed v?

<p>$v^2 / A$ (C)</p> Signup and view all the answers

If the amplitude of the simple harmonic motion of the shadow is 0.50 m, what is the phase constant relative to the x-axis at time t = 0?

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

What is the relationship between circular motion and simple harmonic motion described in the content?

<p>Simple harmonic motion is projected circular motion. (D)</p> Signup and view all the answers

At time t = 0, if the shadow's x-coordinate is 2.00 m and moving to the right, what does this indicate about the object's motion?

<p>The object is in uniform circular motion. (D)</p> Signup and view all the answers

What is the x-coordinate of the shadow as a function of time when the shadow starts at 2.00 m?

<p>$2.00 + A cos(vt)$ (A)</p> Signup and view all the answers

If the object moves with a constant angular speed of 8.00 rad/s in a circular motion, what is the essential feature of this motion?

<p>It is uniform circular motion. (A)</p> Signup and view all the answers

How does the acceleration of the projected point along the x-axis relate to the motion described?

<p>It has the same magnitude as the x component of the radial acceleration. (A)</p> Signup and view all the answers

Flashcards

Simple Harmonic Motion (SHM) Position

The position of an object undergoing simple harmonic motion (SHM) is described by a cosine function of time, where A is the amplitude, v is the angular frequency, and φ is the phase constant. x(t) = A cos(vt + φ)

SHM Velocity

The velocity of an object in simple harmonic motion is given by the derivative of the position function, using the sine function. v(t) = -vA sin(vt + φ)

SHM Acceleration

The acceleration of an object in simple harmonic motion is the derivative of the velocity function, using the cosine function. a(t) = -v²A cos(vt + φ)

Phase Constant (φ)

The phase constant determines the initial position and velocity of the object. It's used to define the initial conditions of the oscillation.

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Initial Condition x(0) & v(0)

The starting position x(0) = A cos(φ) and initial velocity v(0) = -vA sin(φ) of an object in SHM determine the phase constant φ.

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Phase Constant (f)

A constant that describes the starting position and velocity of an object undergoing simple harmonic motion.

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Maximum Speed (vmax)

The fastest speed reached by an object in simple harmonic motion.

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Maximum Acceleration (a max)

The greatest acceleration experienced by an object in simple harmonic motion.

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Simple Harmonic Oscillator (SHO)

A system that oscillates with a restoring force proportional to its displacement from equilibrium.

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Frequency (f) of vibration

The number of oscillations per unit of time for a vibrating object.

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Force Constant (k)

A measure of how stiff an object is, relating force and displacement in a spring.

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Position (x) of SHO

The location of the object in simple harmonic motion at a specific time.

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Velocity (v) of SHO

The rate of change of position of the object in simple harmonic motion.

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Simple Harmonic Motion

A type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium.

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Restoring Force

A force that always acts to return an object to its equilibrium position.

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Amplitude (A)

Maximum displacement from equilibrium position during SHM.

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Angular Frequency (v)

Rate of oscillation in radians per second for SHM.

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Position in SHM

Describes the location of an object undergoing SHM over time.

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Velocity in SHM

Describes how position in an object undergoing SHM changes over time

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Acceleration in SHM

Describes how velocity changes in an object undergoing SHM over time

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Kinetic Energy

The energy of motion

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Potential Energy

The energy due to position or configuration.

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Total Energy

The sum of kinetic and potential energies in a system.

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Circular Motion

Motion along a circular path at a constant speed.

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Oscillatory Motion

Repetitive back-and-forth motion about a central point.

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Treadle Motion

Up-and-down motion of a sewing machine treadle.

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Drive Wheel

Wheel transmitting motion in a sewing machine.

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Simple Harmonic Motion (SHM)

Motion described by a back-and-forth movement; a type of oscillatory motion.

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Reference Circle

Circle used to analyze SHM.

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Angular Speed

Constant rate of rotation.

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Relationship between SHM and Circular Motion

SHM can be viewed as the projection of uniform circular motion onto a straight line.

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Uniform Circular Motion

Motion in a circle with constant speed.

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Simple Harmonic Motion

Back-and-forth motion around an equilibrium point.

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Projected Shadow

The shadow of an object in circular motion, which appears to move in simple harmonic motion.

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Amplitude (SHM)

Maximum displacement from the equilibrium position.

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Phase Constant

Describes the starting position and direction in simple harmonic motion.

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Angular Speed (v)

Rate of change of angle in circular motion.

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Radius (A)

Distance from the center of the circle to the particle.

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X-coordinate (SHM)

Position along an x-axis of a projected shadow during SHM.

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Torsional Pendulum

A rigid object suspended by a wire that exerts a restoring torque proportional to its angular position.

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Torsion Constant (k)

A constant representing the stiffness of the wire in a torsional pendulum, relating torque to angular position

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Period of a Torsional Pendulum

The time it takes for one complete oscillation of the pendulum's angular position.

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Damped Oscillations

Oscillations where the mechanical energy decreases over time due to nonconservative forces (like friction).

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Restoring Torque

The torque that acts to return an object to its equilibrium position in a torsional pendulum.

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Equation of Motion (Torsional)

The equation that describes how the angular position changes over time in a torsional pendulum (2nd order differential equation).

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Simple Harmonic Oscillator (SHO)

A system that oscillates with a restoring force proportional to its displacement from equilibrium.

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Nonconservative Force

A force that causes a loss of mechanical energy in the system.

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Study Notes

Oscillatory Motion

  • Periodic motion is motion of an object that regularly returns to a given position after a fixed time interval.
  • Examples include: a car returning to the driveway, a chandelier swinging, the Earth orbiting the Sun.
  • Other examples include: molecules in solids oscillating, light waves, electromagnetic waves, alternating-current electrical circuits.

Simple Harmonic Motion

  • In simple harmonic motion, the force acting on an object is proportional to its position relative to an equilibrium position and directed opposite the displacement.
  • The acceleration of the object is proportional to its position and directed opposite the displacement from equilibrium.
  • If a block is displaced to a position 'x', the spring exerts a force 'F = -kx' (Hooke's Law) towards equilibrium position.

Analysis Model: Particle in Simple Harmonic Motion

  • The motion is represented by the differential equation: d²x/dt² = -ω²x where ω² = k/m.
  • A solution to the equation is: x(t) = A cos(ωt + φ) where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
  • The period of the motion (T) is related to the angular frequency by T = 2π/ω.
  • The frequency (f) is the inverse of the period: f = 1/T = ω/2π .
  • The amplitude, A, is the maximum displacement from the equilibrium position.
  • The phase constant, φ, depends on the initial position and velocity. For example, if x=A at t = 0, then φ = 0

Mechanical Energy

  • The total mechanical energy (E) of a simple harmonic oscillator is constant and given by: E = kA².
  • Energy is continually transformed between Kinetic (K) and Potential (U) energy forms.
  • Kinetic energy at maximum = E
  • Potential energy at maximum = E

The Pendulum

  • The simple pendulum is a mechanical system of a particle suspended from a fixed point by a string of length L.
  • With small angles of oscillation (𝜃), the motion can be modeled as simple harmonic motion.
  • Period of a simple pendulum: T = 2π√(L/g). T depends only on length (L) and acceleration due to gravity (g).

Damped Oscillations

  • In real-world systems, nonconservative forces (e.g., friction, air resistance) retard the motion.
  • This causes energy to be dissipated, causing the amplitude to decrease over time in an exponential manner.
  • The solution equation is: x = Ae^(-b/2m)t cos(wt + φ).
    • 'A' is the amplitude,
    • 'b' is the damping coefficient,
    • 'm' is the mass,
    • 'ω' is the angular frequency.

Forced Oscillations

  • An external force that varies periodically, like F(t) = Fo sin wt, can compensate for the energy loss due to damping.
  • The amplitude of the oscillation is constant when the energy input per cycle equals the energy dissipated per cycle.
  • When the frequency (ω) of the driving force is close to the natural frequency (ω₀) of the oscillator, resonance occurs, and the amplitude is large.

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