motion of a mass at the end of a spring differential equation for simple harmonic oscillation...
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• Motion of a mass at the end of a spring
• Differential equation for simple harmonic oscillation
• Amplitude, period, frequency and angular frequency
• Energetics
Lecture 23: Simple Harmonic Motion
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Mass at the end of a spring
Mass connected to a spring with spring constant on a frictionless surface
𝐹 𝑥=−𝑘𝑥
Linear restoring spring force
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Spring force
Spring force: stretch or compression force constant
is negative if is positive (stretched spring)
is positive if is negative (compressed spring)
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Differential equation of a SHO
:
−𝑘𝑥=𝑚 𝑑2𝑥𝑑𝑡 2
−𝑘𝑚𝑥= 𝑑2𝑥
𝑑𝑡 2
𝑑2𝑥𝑑𝑡2
=−ω2𝑥 * 𝜔=√ 𝑘𝑚
Differential equation of a Simple Harmonic Oscillator
*We can always write it like this because m and k are positive
Angular frequency
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Solution
and : two “constants of integration” from solution of a second-order differential equation.Determined by the initial conditions.
𝑑2𝑥𝑑𝑡2
=−ω2𝑥 Equation for SHO
General solution:
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Amplitude
A = Amplitude of the oscillation
Range of cosine function: -1…+1 ⟹
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Phase Constant
If φ=0:
To describe motion with different starting points:Add phase constant to shift the cosine function
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Initial conditions
))
→ two equations for and
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Position and velocity
At time :
⟹ Mass stops and reverses direction when it reaches maximum displacement (turning point)
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Simulation
http://www.walter-fendt.de/ph14e/springpendulum.htm
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Period and angular frequency
Time for one complete cycle: period
changes by in time
⟹
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Effect of mass and amplitude on period
⟹
𝜔=√ 𝑘𝑚
⟹ 𝑇=2𝜋 √𝑚𝑘
Demo:Vertical springs showing effect of and
Amplitude does not appear – no effect on period
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Energy in SHO
Potential energy of spring force:
𝐸=½𝑘 𝐴2
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Kinetic and potential energy in SHO
𝐸=𝐾𝑚𝑎𝑥 𝑠𝑖𝑛𝟐(𝜔𝑡+𝜑 )+𝑈𝑚𝑎𝑥𝑐𝑜𝑠𝟐(𝜔𝑡+𝜑)
http://www.walter-fendt.de/ph14e/springpendulum.htm
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Example
A block of mass M is attached to a spring and executes simple harmonic motion of amplitude A. At what displacement(s) x from equilibrium does its kinetic energy equal twice its potential energy?