Intro to Simple Harmonic Motion (Horizontal Springs) Video Lessons

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Problem: Learning Goal:Understand how to determine the constants in the general equation for simple harmonic motion, in terms of given initial conditions.A common problem in physics is to match the particular initial conditions - generally given as an initial position x0 and velocity v) at t=0 - once you have obtained the general solution. You have dealt with this problem in kinematics, where the formulax(t)=x0+v0t+(1/2)at2has two arbitrary constants (technically constants of integration that arise when finding the position given that the acceleration is a constant). The constants in this case are the initial position and velocity, so "fitting" the general solution to the initial conditions is very simple.For simple harmonic motion, it is more difficult to fit the initial conditions, which we take to bex0, the position of the oscillator at t=0, andv0, the velocity of the oscillator at t=0.There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:x(t)=Acos(ωt+ϕ) andx(t)=Ccos(ωt)+Ssin(ωt),where A, ϕ, C, and S are constants, ω is the oscillation frequency, and t is time.Although both expressions have two arbitrary constants--parameters that can be adjusted to fit the solution to the initial conditions--Equation 3 is much easier to use to accommodate x0 and v0.(Equation 2 would be appropriate if the initial conditions were specified as the total energy and the time of the first zero crossing, for example.)Part AFind C and S in terms of the initial position and velocity of the oscillator.Give your answers in terms of x0, v0, and ω. Separate your answers with a comma.

FREE Expert Solution

We'll manipulate the following equations on the basis of initial conditions.

x(t)=Acos(ωt+ϕ)
x(t)=Ccos(ωt)+Ssin(ωt)

Derivation rule:

ddtcos(ax)= -a·sin(ax)

The following trigonometric identies are useful:

cos (a + b) = cos a·cos b - sin a·sin b

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Problem Details

Learning Goal:

Understand how to determine the constants in the general equation for simple harmonic motion, in terms of given initial conditions.

A common problem in physics is to match the particular initial conditions - generally given as an initial position x0 and velocity v) at t=0 - once you have obtained the general solution. You have dealt with this problem in kinematics, where the formula

x(t)=x0+v0t+(1/2)at2

has two arbitrary constants (technically constants of integration that arise when finding the position given that the acceleration is a constant). The constants in this case are the initial position and velocity, so "fitting" the general solution to the initial conditions is very simple.

For simple harmonic motion, it is more difficult to fit the initial conditions, which we take to be

x0, the position of the oscillator at t=0, and
v0, the velocity of the oscillator at t=0.

There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

x(t)=Acos(ωt+ϕ) and

x(t)=Ccos(ωt)+Ssin(ωt),

where A, ϕ, C, and S are constants, ω is the oscillation frequency, and t is time.

Although both expressions have two arbitrary constants--parameters that can be adjusted to fit the solution to the initial conditions--Equation 3 is much easier to use to accommodate x0 and v0.(Equation 2 would be appropriate if the initial conditions were specified as the total energy and the time of the first zero crossing, for example.)

Part A

Find C and S in terms of the initial position and velocity of the oscillator.

Give your answers in terms of x0, v0, and ω. Separate your answers with a comma.

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