Answer to Question #96967 in Classical Mechanics for Surej

Question #96967
body of mass 10g is in SHM with a frequency of 40/π Hz and amplitude of 0.2m.
Find :
a) The maximum force acing on the body. b) the maximum velocity and the
maximum acceleration.
c) the acceleration and the velocity at the point 0.1m from the equilibrium position.
1
Expert's answer
2019-10-22T10:28:57-0400

Dependence of the coordinate (displacement relative to the equilibrium position) is given by the expression


x(t)=Asin(ωt+φ)x(t) = A\sin (\omega t + \varphi )

Where ω\omega is a angular frequency, φ\varphi - the initial phase and AA - amplitude. The angular frequency in our case are


ω=2πν=2π40π[Hz]=80[1s]\omega = 2\pi \nu = 2\pi \cdot {{40} \over \pi }[{\rm{Hz}}] = 80[{{\rm{1}} \over {\rm{s}}}]


Part a):

The magnitude of the force can be found by Newton's second law F=md2xdt2\left| {\vec F} \right| = m\left| {{{{d^2}\vec x} \over {d{t^2}}}} \right| . Let's differentiate x(t)x(t) with respect to tt twice


x˙(t)=Aωcos(ωt+φ)\dot x(t) = A\omega \cos (\omega t + \varphi )x¨(t)=Aω2sin(ωt+φ)\ddot x(t) = - A{\omega ^2}\sin (\omega t + \varphi )

Then, we need to maximize the expression F=Amω2sin(ωt+φ)\left| {\vec F} \right| = Am{\omega ^2}\left| {\sin (\omega t + \varphi )} \right|

It's obvious, that the max(sin(ωt+φ))=1\max (\sin (\omega t + \varphi )) = 1 , then

Fmax=Amω2{\left| {\vec F} \right|_{\max }} = Am{\omega ^2}

Let's calculate

Fmax=0.2[m]10103[kg]802[1s2]=12.8[N]{\left| {\vec F} \right|_{\max }} = 0.2[{\rm{m}}] \cdot 10 \cdot {10^{ - 3}}[{\rm{kg}}] \cdot {80^2}[{{\rm{1}} \over {{{\rm{s}}^{\rm{2}}}}}] = 12.8[{\rm{N}}]

Part b):

The maximum acceleration is (Newton's second law)


amax=Fmaxm=Aω2{\left| {\vec a} \right|_{\max }} = {{{{\left| {\vec F} \right|}_{\max }}} \over m} = A{\omega ^2}amax=0.2[m](80)2[1s2]=1280[ms2]{\left| {\vec a} \right|_{\max }} = 0.2[{\rm{m}}] \cdot {(80)^2}[{{\rm{1}} \over {{{\rm{s}}^{\rm{2}}}}}] = 1280[{{\rm{m}} \over {{{\rm{s}}^{\rm{2}}}}}]


The maximum of velocity can be found maximizing the first derivative


v=Aωcos(ωt+φ)\left| {\vec v} \right| = A\omega \left| {\cos (\omega t + \varphi )} \right|

Again, max(cos(ωt+φ))=1\max (\left| {\cos (\omega t + \varphi )} \right|) = 1 and


vmax=Aω{\left| {\vec v} \right|_{\max }} = A\omegavmax=0.2[m]80[1s]=16[ms]{\left| {\vec v} \right|_{\max }} = 0.2[{\rm{m}}] \cdot 80[{{\rm{1}} \over {\rm{s}}}] = 16[{{\rm{m}} \over {\rm{s}}}]


Part c):

We need to find the such time tt that x(t)=0.1[m]x(t)=0.1[{\rm{m}}]. Let φ=0\varphi =0 and then

Then

x(t)A=sin(ωt){{x(t)} \over A} = \sin (\omega t)t=1ωarcsinx(t)At = {1 \over \omega }\arcsin {{x(t)} \over A}

In our case


t=180[1s]arcsin(0.1[m]0.2[m])=180[1s]π6=π480[s]t = {1 \over {80[{{\rm{1}} \over {\rm{s}}}]}}\arcsin ({{0.1[{\rm{m}}]} \over {0.2[{\rm{m}}]}}) = {1 \over {80[{{\rm{1}} \over {\rm{s}}}]}}{\pi \over 6} = {\pi \over {480}}[{\rm{s}}]

Then the velocity at this time can be calculates using the expression for the first derivative


v(π480)=0.2[m]80[1s]cos(80[1s]π480[s])=16[ms]32=83[ms]\left| {v({\pi \over {480}})} \right| = 0.2[{\rm{m}}] \cdot 80[{{\rm{1}} \over {\rm{s}}}]\left| {\cos (80[{{\rm{1}} \over {\rm{s}}}] \cdot {\pi \over {480}}{\rm{[s]}})} \right| = 16[{{\rm{m}} \over {\rm{s}}}] \cdot {{\sqrt 3 } \over 2} = 8\sqrt 3 [{{\rm{m}} \over {\rm{s}}}]

For the acceleration we use the second derivative


a(π480)=0.2[m]802[1s2]sin(80[1s]π480[s])=1280[ms2]12=640[ms]\left| {a({\pi \over {480}})} \right| = \left| { - 0.2[{\rm{m}}] \cdot {{80}^2}[{{\rm{1}} \over {{{\rm{s}}^2}}}]} \right|\left| {\sin (80[{{\rm{1}} \over {\rm{s}}}] \cdot {\pi \over {480}}{\rm{[s]}})} \right| = 1280[{{\rm{m}} \over {{{\rm{s}}^2}}}] \cdot {1 \over 2} = 640[{{\rm{m}} \over {\rm{s}}}]




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