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Simple Harmonic Motion, Mass Spring System - Amplitude, Frequency, Velocity - Physics Problems

The Organic Chemistry Tutor・2 minutes read

Periodic motion involves oscillation and repetition, as seen in examples like the mass-spring system and the simple pendulum, with key concepts like Hooke's Law and calculating maximum velocity and acceleration. Damped harmonic motion, different types of damping, and resonant frequency are factors influencing the behavior of oscillatory systems.

Insights

Periodic motion involves repetitive back-and-forth movement, seen in systems like the mass-spring and pendulum setups.
Hooke's Law states that the restoring force in a spring is proportional to the displacement from equilibrium, with the spring constant determining the force needed to stretch or compress the spring.
Mechanical energy in a system remains constant without friction, with kinetic energy peaking when equal to mechanical energy and potential energy at its maximum when displacement matches the amplitude.

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Recent questions

What is periodic motion?
Motion that repeats or oscillates back and forth.

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Summary

00:00

Understanding Periodic Motion and Hooke's Law

Periodic motion is motion that repeats or oscillates back and forth.
Examples of periodic motion include the mass-spring system and the simple pendulum.
In the mass-spring system, a spring connected to a mass is stretched from its equilibrium position.
The restoring force in the spring system pulls the mass back towards equilibrium when stretched.
Hooke's Law states that the restoring force is equal to negative k times x.
The spring constant, k, is measured in newtons per meter and determines the force needed to stretch or compress the spring.
The negative sign in Hooke's Law indicates the restoring force opposes the direction of motion.
To calculate the force needed to stretch a spring, use the equation f = kx.
The force required to stretch a spring by 25 cm with a spring constant of 300 N/m is 75 N.
The force needed to stretch a spring by 120 cm with a force of 150 N is 600 N, with a spring constant of 500 N/m.

23:24

Maximize Energy and Motion in Springs

Kinetic energy (KE) is at its maximum when it equals the mechanical energy of the system, which remains constant without friction.
Potential energy (PE) is at its maximum when the displacement is equal to the amplitude (a), with KE being zero.
The maximum velocity can be calculated using the equation v = √(k/m) * a.
Maximum acceleration is found by the equation a = k * a / m.
Velocity as a function of displacement (x) can be determined using the equation v = ± √(k/m) * a * √(1 - x^2/a^2).
The frequency of a spring-mass oscillator is inversely related to its period, with frequency being cycles per second (Hertz).
Doubling the maximum displacement quadruples the mechanical energy of the system.
The maximum velocity increases proportionally to the maximum displacement.
The maximum acceleration also increases proportionally to the maximum displacement.
For a spring with a massless block oscillating on a frictionless surface, the maximum velocity and acceleration can be calculated using specific equations.

46:16

Calculating Energy and Velocity in Oscillatory Motion

The equation for maximum velocity is the square root of k divided by m times a, where k is 49, m is 2, and a is 0.2, resulting in a maximum velocity of approximately 0.99 meters per second.
To find the potential energy at 0.10 meters from equilibrium, use the equation 1/2 kx^2 with k as 49 and x as 0.10, yielding a potential energy of 0.245 joules.
Kinetic energy is calculated using 1/2 mv^2, with v as 0.8574 meters per second, resulting in a kinetic energy of 0.735 joules.
Mechanical energy can be found by 1/2 k times the amplitude squared, giving a total energy of 0.98 joules.
Another method to find mechanical energy is by adding potential and kinetic energy, resulting in the same value of 0.98 joules.
The velocity at any point x can be found using the equation v = vmax times the square root of 1 - x^2 divided by a^2, where vmax is 0.99, x is 0.1, and a is 0.2, giving a velocity of 0.8574 meters per second.
The speed of a block released from a compressed spring with a force of 500 newtons and a displacement of 0.35 meters can be calculated by finding the spring constant k as 1429 newtons per meter and using conservation of energy to determine a speed of 26.5 meters per second.
The kinetic energy of the block upon release is 87.53 joules, which can also be calculated using the work equation by finding the area under the force-displacement graph.
The period of a spring's oscillation is 2π times the square root of m divided by k, where increasing mass increases the period and increasing k decreases the period.
Frequency is the reciprocal of the period, with increasing k increasing frequency and increasing mass decreasing frequency due to their impact on acceleration and speed in oscillatory motion.

01:09:04

"Frequency and Spring Constant Calculations Explained"

Increasing the value of k leads to faster oscillations and a higher frequency.
Heavy objects have slower movement and lower frequency when mass is increased.
A 0.25 kg block is attached to a spring stretched 0.25 meters by a 200 N force.
Frequency calculation: 1 / (2π * √(k / m)), k = 200 N / 0.25 m = 800 N/m.
Frequency calculation: 1 / (2π * √(800 / 0.25)) = 9 Hz, Period = 1 / 9 = 0.11 s.
Frequency calculation for a 70 kg person on a 1200 kg car compressing springs by 2 cm.
Spring constant k calculation: 70 * 9.8 / 0.02 = 34,000 N/m.
Frequency calculation: 1 / (2π * √(34,300 / 1270)) = 0.83 Hz.
Finding spring constant k for an insect in a spider web using frequency and mass.
Spring constant k calculation: 3.95 N/m, Frequency calculation: 31.6 Hz.
Frequency comparison for a block attached to springs with different constants: 23.7 Hz for 500 N/m.

01:30:23

Vibrating Mass Dynamics: Velocity, Acceleration, Energy

The velocity as a function of time is v max times the square root of m divided by k times negative sine 2 pi f t.
The instantaneous velocity is equal to the maximum velocity, represented as negative v max times sine 2 pi f t.
The acceleration is the derivative of the velocity function, given by negative k over m times a cosine 2 pi f t.
The maximum acceleration is k times the amplitude divided by m, leading to the acceleration function as a max times cosine 2 pi f t.
The amplitude of a 0.75 kg mass vibrating is 0.6 meters, with a frequency of about 1.464 Hz and a spring constant of 63.5 N/m.
The total energy of the system is 11.42 Joules, with potential energy at 0.6 displacement being 11.42 Joules.
At x=0, kinetic energy equals total energy (11.42 J) and potential energy is zero; at x=0.6, kinetic energy is zero and potential energy is 11.42 J.
The potential energy at x=0.2 is 1.27 J, with kinetic energy at that point being 10.15 J.
For a 0.55 kg block vibrating, the maximum velocity is 18.6 m/s, and the maximum acceleration is 230.64 m/s^2.
At 0.5 meters from equilibrium, the velocity is 1.545 m/s, and the acceleration is -229.84 m/s^2, leading to a restoring force of -126.4 N.

01:53:28

Harmonic Motion and Damping in Physics

Motion in the positive y direction is represented by a positive sine function, while motion in the negative y direction is represented by a negative sine function.
The position function for a mass with an amplitude of 0.45 meters is determined to be a sine 29.3t.
Starting below the equilibrium point results in motion described by a negative cosine function, specifically -0.45 cosine 29.28t.
Damped harmonic motion occurs when friction is present, leading to a decrease in amplitude over time, with three types of damping: underdamped, overdamped, and critically damped. Resonant frequency occurs when an applied force matches the natural frequency of an object, leading to increased amplitude.

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