In the absence of oscillations, equilibrium between the force of gravity and the restoring force takes place,"{{F}_{r}}={{F}_{grav}}" or "kx=mg" (it is assumed that the spring is directed along the x axis). This gives the extension of the spring
"{{x}_{0}}=\\frac{mg}{k}=\\frac{0.2\\,kg\\cdot 10\\,m\\cdot {{s}^{-2}}}{100\\,N\\cdot {{m}^{-1}}}=0.02\\,m" 1) In order to obtain the equation of motion of the pendulum we write Newton's second law
"ma={{F}_{el}}+{{F}_{fr}}"where "a" is the acceleration, "a=\\frac{{{d}^{2}}x}{d{{t}^{2}}}" ; "{{F}_{r}}" is the restoring force, "{{F}_{r}}=-kx" ; "{{F}_{fr}}" is the frictional force, "{{F}_{fr}}=-b\\cdot v=-b\\frac{dx}{dt}" "(b=6\\,N\\cdot s\\cdot {{m}^{-1}})" . Then the equation of motion of the pendulum is
"m\\frac{{{d}^{2}}x}{d{{t}^{2}}}=-kx-b\\frac{dx}{dt}"
or
"\\frac{{{d}^{2}}x}{d{{t}^{2}}}+\\frac{b}{m}\\frac{dx}{dt}+\\frac{k}{m}x=0" Solve the equation and find the period of oscillations. To find x we substitute into equation the solution as exponential function "x={{e}^{\\lambda t}}" . We get
"{{\\lambda }^{2}}{{e}^{\\lambda t}}+\\frac{b}{m}\\lambda {{e}^{\\lambda t}}+\\frac{k}{m}{{e}^{\\lambda t}}=0"
or
"\\left( {{\\lambda }^{2}}+\\frac{b}{m}\\lambda +\\frac{k}{m} \\right){{e}^{\\lambda t}}=0"
Then we get the auxiliary equation
"{{\\lambda }^{2}}+\\frac{b}{m}\\lambda +\\frac{k}{m}=0"
The solution of this equation is
"\\lambda =-\\frac{b}{2m}\\pm \\sqrt{\\frac{{{b}^{2}}}{4{{m}^{2}}}-\\frac{k}{m}}=-\\frac{b}{2m}\\pm i\\sqrt{\\frac{k}{m}-\\frac{{{b}^{2}}}{4{{m}^{2}}}}"
where "\\sqrt{\\frac{k}{m}-\\frac{{{b}^{2}}}{4{{m}^{2}}}}=\\omega" is the angular frequency. The period of free oscillations is
"T=\\frac{2\\pi }{\\omega }=\\frac{2\\pi }{\\sqrt{\\frac{k}{m}-\\frac{{{b}^{2}}}{4{{m}^{2}}}}}"Substituting known values "m=0.2\\,kg,\\,\\,\\,k=100\\,N\\cdot {{m}^{-1}},\\,\\,\\,b=6\\,N\\cdot s\\cdot {{m}^{-1}}" , we get "T\\approx 0.38\\,s"
2) If a harmonic force F = 20 cos 20t= F0 cos Ωt is applied, then Newton's second law is
"m\\frac{{{d}^{2}}x}{d{{t}^{2}}}=-kx-b\\frac{dx}{dt}+F"Then the equation of motion of the pendulum is
"\\frac{{{d}^{2}}x}{d{{t}^{2}}}+\\frac{b}{m}\\frac{dx}{dt}+\\frac{k}{m}x=\\frac{{{F}_{0}}}{m}\\cos \\Omega t"
or
"\\frac{{{d}^{2}}x}{d{{t}^{2}}}+\\alpha \\frac{dx}{dt}+\\omega _{0}^{2}x={{A}_{0}}\\cos \\Omega t"
where "\\alpha =b\/m,\\,\\,\\,\\omega _{0}^{2}=b\/m,\\,\\,\\,{{A}_{0}}={{F}_{0}}\/m" are denoted.
To calculate the amplitude of forced oscillations we find the particular solution of this equation. We seek a particular solution of the form
"{{x}_{p}}\\left( t \\right)=A\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t+B\\text{sin }\\!\\!\\Omega\\!\\!\\text{ }t"
where A and B are unknowing constants which we now find. Substitute "{{x}_{p}}\\left( t \\right)" into equation
"-A{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t-B{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}\\text{sin }\\!\\!\\Omega\\!\\!\\text{ }t+\\alpha \\left( -A\\text{ }\\!\\!\\Omega\\!\\!\\text{ sin }\\!\\!\\Omega\\!\\!\\text{ }t+B\\text{ }\\!\\!\\Omega\\!\\!\\text{ cos }\\!\\!\\Omega\\!\\!\\text{ }t \\right)"
"+\\omega _{0}^{2}\\left( A\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t+B\\text{sin }\\!\\!\\Omega\\!\\!\\text{ }t \\right)={{A}_{0}}\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t" or
"\\left( -A{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}+B\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ }+A\\omega _{0}^{2} \\right)\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t"
"+\\left( -B{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}-A\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ }+B\\omega _{0}^{2} \\right)\\text{sin }\\!\\!\\Omega\\!\\!\\text{ }t={{A}_{0}}\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t" This is true if
"-A{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}+B\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ }+A\\omega _{0}^{2}={{A}_{0}}" and
"-B{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}-A\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ }+B\\omega _{0}^{2}=0" Solving this system of equation we get
"A=\\frac{{{A}_{0}}\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}"
"B=\\frac{{{A}_{0}}\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ }}{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}" Then the particular solution is
"{{x}_{p}}\\left( t \\right)={{A}_{0}}\\left( \\frac{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)\\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t}{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}+\\frac{\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ sin }\\!\\!\\Omega\\!\\!\\text{ }t}{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}} \\right)" Denote
"\\text{cos}\\varphi =\\frac{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}{\\sqrt{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}},\\text{ }\\!\\!~\\!\\!\\text{ }\\!\\!~\\!\\!\\text{ }\\!\\!~\\!\\!\\text{ }\\!\\!~\\!\\!\\text{ sin}\\varphi =\\frac{\\alpha \\text{ }\\!\\!\\Omega\\!\\!\\text{ }}{\\sqrt{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}}"
then
"{{x}_{p}}\\left( t \\right)=\\frac{{{A}_{0}}}{\\sqrt{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}}\\left( \\text{cos}\\varphi \\text{cos }\\!\\!\\Omega\\!\\!\\text{ }t+\\text{sin}\\varphi \\text{sin }\\!\\!\\Omega\\!\\!\\text{ }t \\right)"
or
"{{x}_{p}}\\left( t \\right)=\\frac{{{A}_{0}}}{\\sqrt{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}}\\text{cos}\\left( \\text{ }\\!\\!\\Omega\\!\\!\\text{ }t-\\varphi \\right)"
The amplitude of forced oscillations is
"{{x}_{f}}=\\frac{{{A}_{0}}}{\\sqrt{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}}"
Substituting Ω=20,
"{{A}_{0}}=\\frac{{{F}_{0}}}{m}=\\frac{20}{0.2}=100,\\,\\,\\,\\omega _{0}^{2}=\\frac{k}{m}=\\frac{100}{0.2}=500"
"\\alpha =\\frac{b}{m}=\\frac{6}{0.2}=30"
we get
"{{x}_{f}}=\\frac{{{A}_{0}}}{\\sqrt{{{\\left( \\omega _{0}^{2}-{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}} \\right)}^{2}}+{{\\alpha }^{2}}{{\\text{ }\\!\\!\\Omega\\!\\!\\text{ }}^{2}}}}=\\frac{100}{\\sqrt{{{\\left( 500-400 \\right)}^{2}}+900\\cdot 400}}\\approx 0.164\\,meters"
Answer:
the equation of motion of the pendulum is
"\\frac{{{d}^{2}}x}{d{{t}^{2}}}+\\frac{b}{m}\\frac{dx}{dt}+\\frac{k}{m}x=0"
the period of free oscillations is "T\\approx 0.38\\,s"
the amplitude of forced oscillations is "{{x}_{f}}\\approx 0.164\\,meters"
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