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পর্যাবৃত্ত গতি || Engineering Series class 2

Apar's Classroom・2 minutes read

The text discusses complex mathematical equations and concepts related to simple harmonic motion, buoyancy, energy, and oscillations. It covers formulas for velocity, acceleration, potential energy, and kinetic energy, emphasizing the relationship between variables in these physics concepts.

Insights

  • Differentiation process leads to the velocity equation V = Omega * sqrt(a^2 - y^2), with maximum velocity at equilibrium position.
  • Acceleration equation derived from velocity equation: A = -A * Omega^2 * sin(Omega), with maximum acceleration only applicable from equilibrium position.
  • Understanding Archimedes' principle and buoyancy force in submerged objects, highlighting the importance of displacement and weight differences.
  • Analysis of total energy in simple harmonic motion, emphasizing the constant nature of total energy, and exploring the relationship between kinetic and potential energy.

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Recent questions

  • What is the relationship between velocity and displacement equations?

    In simple oscillating motion, the velocity equation V = Omega * sqrt(a^2 - y^2) and the displacement equation y = a sin omega t are interconnected. The velocity equation determines the speed of the object at any given time, while the displacement equation represents the position of the object at that time. The maximum velocity occurs at the equilibrium position, where the displacement is zero. Understanding this relationship helps in visualizing how the object moves in simple harmonic motion, with the velocity and displacement equations complementing each other to describe the complete motion cycle.

  • How is acceleration calculated in simple harmonic motion?

    The acceleration in simple harmonic motion is derived from the differentiation of the velocity equation. The formula for acceleration is A = -A * Omega^2 * sin(Omega), where A represents the amplitude of the motion and Omega is the angular frequency. Maximum acceleration only occurs when the motion starts from the equilibrium position, where the displacement is zero. By understanding the acceleration equation, one can determine how the object's speed changes over time and the direction of the acceleration at different points in the motion cycle.

  • What is Archimedes' principle and its application in buoyancy?

    Archimedes' principle states that the weight of an object in water is equal to the weight of the liquid displaced by the object. When an object is submerged in water, it experiences an upward force known as buoyancy, which is equal to the weight of the liquid displaced. By calculating the difference between the object's weight in air and water, one can determine the weight lost by the object in water. Understanding this principle is crucial for various applications, such as determining the mass of liquid displaced by a submerged object and analyzing the forces acting on objects in fluid environments.

  • How is total energy calculated in simple harmonic motion?

    In simple harmonic motion, the total energy of the system remains constant regardless of time or position. The total energy is the sum of kinetic and potential energy, where kinetic energy is calculated using the formula EK = 1/2 MV^2 and potential energy is determined based on the position of the object in the motion cycle. By understanding the relationship between kinetic and potential energy, one can grasp how energy is transferred and conserved throughout the oscillation process. The graph of energy vs. position remains constant, reflecting the equilibrium state of the system's energy.

  • What is the formula for the period of oscillation in a spring system?

    The formula for the period of oscillation in a spring system is T = 2π√(m/k), where T represents the period, m is the mass of the object attached to the spring, and k is the spring constant. This formula is applicable to various types of springs, whether hanging or on a plane, and provides a way to calculate the time taken for one complete oscillation cycle. Understanding this formula helps in predicting the behavior of the spring system and determining the frequency of oscillation based on the mass and spring constant values.

Related videos

Summary

00:00

Understanding Simple Oscillating Motion Equations

  • Request for feedback on voice and text clarity
  • Introduction to chapter two of the motions being discussed
  • Previous class focused on number class and theoretical discussions
  • Today's class will delve into mathematical terms and interesting math
  • Explanation of the relationship y = a sin omega t from the last class
  • Definition of terms like y, a, and omega in the equation
  • Differentiation process to find the velocity equation
  • Formula for velocity equation: V = Omega * sqrt(a^2 - y^2)
  • Maximum velocity occurs at equilibrium position
  • Acceleration equation derived from differentiation of velocity equation
  • Formula for acceleration: A = -A * Omega^2 * sin(Omega)
  • Maximum acceleration only applicable when motion starts from equilibrium position
  • Calculation of period in a given problem involving velocity and acceleration ratios
  • Explanation of the difference between velocity and displacement equations
  • Calculation of angle in displacement equation: pi/2 radians
  • Relationship between velocity and displacement equations in simple oscillating motion

16:22

Coordinate Shifting and Simple Harmonic Motion

  • Minus signifies withdrawal in coordinate shifting
  • Sin 180 + θ = - sin θ, where 180 equals π + θ
  • Differentiation results in pi plus omega t minus omega t
  • The diffusion equation involves a particle's oscillating motion
  • The particle's velocity decreases from maximum to zero
  • Tripling the velocity maintains simple harmonic motion
  • The equation for velocity is V = Omega root over A squared minus x squared
  • The new expansion results in a distance of 7 by 3
  • The graph of velocity vs. time shows peaks at omega and zero
  • Acceleration graph follows a similar process, with peaks at omega and zero

32:23

Graphing Acceleration and Velocity in Motion

  • The maximum acceleration on one side is Omega Square, while the maximum acceleration in the opposite direction is -Omega Square.
  • Setting the value of T to zero results in the acceleration being zero.
  • Two points are identified as (0,0) and (T/4, 1).
  • The graph of X is discussed, showing the difference between a sine graph and a cosine graph.
  • The process of graphing any relation is explained, emphasizing the importance of understanding the differences between graphs.
  • Practical examples are given to illustrate how to graph equations and interpret the results.
  • The equation for X is provided, leading to the need to draw an acceleration versus time graph.
  • The process of calculating the velocity of a particle at a specific time is detailed using the formula V = Omega root over A square.
  • The acceleration of a particle 3 cm away from its position is calculated as -Omega square times 0.03.
  • A practical math problem is solved step by step, demonstrating the application of formulas and calculations to find the position of an object.

48:26

Key Points: Plus or Minus in Equations

  • Writing down key points is crucial for understanding when to take plus or minus in mathematical equations.
  • Taking the value of X as plus or minus is mandatory in certain equations.
  • Plus or minus is only applicable for X and not for other variables like V.
  • When dealing with quadratic equations, determining the correct conditions for plus or minus is essential.
  • The acceleration of an object can be calculated based on its mass and oscillation frequency.
  • Hooke's law states that force is proportional to displacement in simple harmonic motion.
  • Calculating the acceleration of a particle involves using the formula A = -Omega^2X.
  • Conversion of units from centimeters to meters is necessary for accurate calculations.
  • Solving equations involving multiple variables requires careful consideration and calculation.
  • Understanding Archimedes' principle involves determining the weight lost by an object when submerged in water.

01:05:04

Buoyancy and Archimedes' Principle in Practice

  • Understanding the concept of buoyancy: When an object is submerged in water, it experiences an upward force due to water pushing upwards and its weight pulling downwards.
  • Buoyancy force and weight: The upward thrust of water is known as buoyancy, which is equal to the weight of the object lost in water.
  • Calculation of lost weight: The weight lost by an object in water is equal to the difference between its weight in air and water.
  • Archimedes' principle: Archimedes provided the concept that the weight of an object in water is equal to the weight of the liquid displaced by the object.
  • Volume of liquid displaced: The volume of liquid removed by an object when submerged is equal to the volume of liquid displaced, which can be calculated using density and volume.
  • Special case of floating objects: When an object floats, its weight in water is zero, leading to the buoyancy force being equal to the weight of the object in air.
  • Application of formulas: The volume of liquid displaced by a submerged object can be calculated by determining the volume of the object submerged in water.
  • Understanding buoyancy in practice: Visualizing the relationship between the volume of liquid displaced and the buoyancy force helps in comprehending the concept.
  • Calculation of mass of liquid removed: By using the formula mass = density x volume, the mass of liquid removed by a submerged object can be determined.
  • Practical application of concepts: Applying the principles of buoyancy and Archimedes' principle allows for the calculation of the mass of liquid displaced by an object in water.

01:21:30

"Mass, Motion, Buoyancy: Physics Explained"

  • Pylum obtained the mass of the entire object
  • Got the mass of the whole object
  • Extracting data on the mass of the object
  • Explaining the concept of giving motion to the object
  • Describing the process of giving circular motion
  • Understanding the buoyancy force and its impact
  • Calculating the buoyancy force and its direction
  • Determining the formula for positive force
  • Solving for the frequency of the oscillator
  • Analyzing the volume and density of a solid sphere in liquid

01:39:28

Buoyancy, Harmonic Motion, and Energy Analysis

  • The text discusses the concept of buoyancy and simple harmonic motion.
  • It mentions the volume of the part submerged and the extra dip taken.
  • The buoyancy force created by the displaced water is highlighted.
  • The text delves into converting to simple harmonic motion and the formula for buoyancy force.
  • It emphasizes the importance of the cross-sectional area and the volume of the object.
  • The formula for determining the volume of a sphere is detailed.
  • The text explains the mass of the object and the frequency needed for extraction.
  • It discusses the energy and potential energy in simple harmonic motion.
  • The maximum value of potential energy and the energy graph are explored.
  • The text concludes with a discussion on the potential energy vs. time graph and the force vs. time graph.

01:55:39

Analyzing Equilibrium Through Energy Graphs

  • The object in equilibrium is analyzed through memory vs. time graph.
  • Energy vs. time graph is examined to determine potential energy's maximum value at half k.
  • The process involves cutting spots before graphing and understanding the relationship between energy and time.
  • The graph is drawn with specific values, such as t/4 and 3t/4, to represent different points.
  • The relationship between y and u is explored, with y=0 resulting in u=0.
  • The concept of maximum values and directions in relation to y and u is clarified.
  • The repetition of motion and energy is discussed, emphasizing the periodicity of energy.
  • The formula for kinetic energy (EK = 1/2 MV^2) is explained, with details on velocity and momentum.
  • The graph of kinetic energy vs. memory and kinetic energy vs. time is analyzed.
  • The relationship between kinetic energy, velocity, and time is explored, highlighting the proportional nature of their values.

02:11:31

"Energy formulas and constants in motion"

  • Particle motion frequencies are the same when multiplied by pi.
  • Multiplying by pi converts energy to omega power and speed to omega.
  • Total energy is the sum of kinetic and potential energy.
  • The formula for kinetic energy is half of a square.
  • The total energy formula is the sum of kinetic and potential energy formulas.
  • Total energy remains constant regardless of time or position.
  • The pendulum's total energy varies at different positions during oscillation.
  • The graph of energy vs. position remains constant.
  • The formula for total energy is derived by adding kinetic and potential energy formulas.
  • The formula for total energy is calculated using the spring constant and mass.

02:43:29

Spring Oscillation Formulas and Calculations

  • Formula for spring oscillation: T = 2π√(m/k)
  • Applicable to all types of springs: hanging or on a plane
  • Specific formula for hanging spring: T = 2π√(m/k)
  • Alternative formula for suspension springs: T = 2π√(e/G)
  • Expansion due to hanging mass: spring elongates
  • Equilibrium state of expansion: X remains constant
  • Formula for damped spring: T = 2π√(e/G)
  • Calculation example: mass of 100g, spring constant of 25 N/m
  • Period of oscillation calculation: T = 3.974 seconds
  • Frequency calculation: f = 0.252 Hz

03:01:10

"Deriving Maximum Velocity Formula for Colliding Masses"

  • The formula for f prime by f is the square root of capital M divided by capital M plus small M.
  • The initial state involves a spring with mass capital M swinging until it reaches the middle position.
  • When a new mass small M is added, the spring continues to oscillate with a new frequency.
  • Before the collision, the momentum of the mass is capital M times velocity V.
  • The maximum velocity occurs just before the collision when the mass is at equilibrium.
  • After the collision, the mass becomes capital M plus small M, with a new maximum velocity V prime.
  • The formula for maximum velocity V prime is derived by dividing Omega by Omega prime, resulting in f by f prime.
  • The final answer is obtained by calculating the square root of capital M divided by capital M plus small M multiplied by the initial formula of f prime by f.
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