Dependence of the coordinate (displacement relative to the equilibrium position) is given by the expression
x ( t ) = A sin ( ω t + φ ) x(t) = A\sin (\omega t + \varphi ) x ( t ) = A sin ( ω t + φ ) Where ω \omega ω is a angular frequency, φ \varphi φ - the initial phase and A A A - amplitude. The angular frequency in our case are
ω = 2 π ν = 2 π ⋅ 40 π [ H z ] = 80 [ 1 s ] \omega = 2\pi \nu = 2\pi \cdot {{40} \over \pi }[{\rm{Hz}}] = 80[{{\rm{1}} \over {\rm{s}}}] ω = 2 π ν = 2 π ⋅ π 40 [ Hz ] = 80 [ s 1 ]
Part a):
The magnitude of the force can be found by Newton's second law ∣ F ⃗ ∣ = m ∣ d 2 x ⃗ d t 2 ∣ \left| {\vec F} \right| = m\left| {{{{d^2}\vec x} \over {d{t^2}}}} \right| ∣ ∣ F ∣ ∣ = m ∣ ∣ d t 2 d 2 x ∣ ∣ . Let's differentiate x ( t ) x(t) x ( t ) with respect to t t t twice
x ˙ ( t ) = A ω cos ( ω t + φ ) \dot x(t) = A\omega \cos (\omega t + \varphi ) x ˙ ( t ) = A ω cos ( ω t + φ ) x ¨ ( t ) = − A ω 2 sin ( ω t + φ ) \ddot x(t) = - A{\omega ^2}\sin (\omega t + \varphi ) x ¨ ( t ) = − A ω 2 sin ( ω t + φ ) Then, we need to maximize the expression ∣ F ⃗ ∣ = A m ω 2 ∣ sin ( ω t + φ ) ∣ \left| {\vec F} \right| = Am{\omega ^2}\left| {\sin (\omega t + \varphi )} \right| ∣ ∣ F ∣ ∣ = A m ω 2 ∣ sin ( ω t + φ ) ∣
It's obvious, that the max ( sin ( ω t + φ ) ) = 1 \max (\sin (\omega t + \varphi )) = 1 max ( sin ( ω t + φ )) = 1 , then
∣ F ⃗ ∣ max = A m ω 2 {\left| {\vec F} \right|_{\max }} = Am{\omega ^2} ∣ ∣ F ∣ ∣ m a x = A m ω 2 Let's calculate
∣ F ⃗ ∣ max = 0.2 [ m ] ⋅ 10 ⋅ 1 0 − 3 [ k g ] ⋅ 8 0 2 [ 1 s 2 ] = 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}}] ∣ ∣ F ∣ ∣ m a x = 0.2 [ m ] ⋅ 10 ⋅ 1 0 − 3 [ kg ] ⋅ 8 0 2 [ s 2 1 ] = 12.8 [ N ]
Part b):
The maximum acceleration is (Newton's second law)
∣ a ⃗ ∣ max = ∣ F ⃗ ∣ max m = A ω 2 {\left| {\vec a} \right|_{\max }} = {{{{\left| {\vec F} \right|}_{\max }}} \over m} = A{\omega ^2} ∣ a ∣ m a x = m ∣ F ∣ m a x = A ω 2 ∣ a ⃗ ∣ max = 0.2 [ m ] ⋅ ( 80 ) 2 [ 1 s 2 ] = 1280 [ m s 2 ] {\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}}}}}] ∣ a ∣ m a x = 0.2 [ m ] ⋅ ( 80 ) 2 [ s 2 1 ] = 1280 [ s 2 m ]
The maximum of velocity can be found maximizing the first derivative
∣ v ⃗ ∣ = A ω ∣ cos ( ω t + φ ) ∣ \left| {\vec v} \right| = A\omega \left| {\cos (\omega t + \varphi )} \right| ∣ v ∣ = A ω ∣ cos ( ω t + φ ) ∣ Again, max ( ∣ cos ( ω t + φ ) ∣ ) = 1 \max (\left| {\cos (\omega t + \varphi )} \right|) = 1 max ( ∣ cos ( ω t + φ ) ∣ ) = 1 and
∣ v ⃗ ∣ max = A ω {\left| {\vec v} \right|_{\max }} = A\omega ∣ v ∣ m a x = A ω ∣ v ⃗ ∣ max = 0.2 [ m ] ⋅ 80 [ 1 s ] = 16 [ m s ] {\left| {\vec v} \right|_{\max }} = 0.2[{\rm{m}}] \cdot 80[{{\rm{1}} \over {\rm{s}}}] = 16[{{\rm{m}} \over {\rm{s}}}] ∣ v ∣ m a x = 0.2 [ m ] ⋅ 80 [ s 1 ] = 16 [ s m ]
Part c):
We need to find the such time t t t that x ( t ) = 0.1 [ m ] x(t)=0.1[{\rm{m}}] x ( t ) = 0.1 [ m ] . Let φ = 0 \varphi =0 φ = 0 and then
Then
x ( t ) A = sin ( ω t ) {{x(t)} \over A} = \sin (\omega t) A x ( t ) = sin ( ω t ) t = 1 ω arcsin x ( t ) A t = {1 \over \omega }\arcsin {{x(t)} \over A} t = ω 1 arcsin A x ( t ) In our case
t = 1 80 [ 1 s ] arcsin ( 0.1 [ m ] 0.2 [ m ] ) = 1 80 [ 1 s ] π 6 = π 480 [ 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}}] t = 80 [ s 1 ] 1 arcsin ( 0.2 [ m ] 0.1 [ m ] ) = 80 [ s 1 ] 1 6 π = 480 π [ s ] Then the velocity at this time can be calculates using the expression for the first derivative
∣ v ( π 480 ) ∣ = 0.2 [ m ] ⋅ 80 [ 1 s ] ∣ cos ( 80 [ 1 s ] ⋅ π 480 [ s ] ) ∣ = 16 [ m s ] ⋅ 3 2 = 8 3 [ m s ] \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}}}] ∣ ∣ v ( 480 π ) ∣ ∣ = 0.2 [ m ] ⋅ 80 [ s 1 ] ∣ ∣ cos ( 80 [ s 1 ] ⋅ 480 π [ s ] ) ∣ ∣ = 16 [ s m ] ⋅ 2 3 = 8 3 [ s m ] For the acceleration we use the second derivative
∣ a ( π 480 ) ∣ = ∣ − 0.2 [ m ] ⋅ 80 2 [ 1 s 2 ] ∣ ∣ sin ( 80 [ 1 s ] ⋅ π 480 [ s ] ) ∣ = 1280 [ m s 2 ] ⋅ 1 2 = 640 [ m s ] \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}}}] ∣ ∣ a ( 480 π ) ∣ ∣ = ∣ ∣ − 0.2 [ m ] ⋅ 80 2 [ s 2 1 ] ∣ ∣ ∣ ∣ sin ( 80 [ s 1 ] ⋅ 480 π [ s ] ) ∣ ∣ = 1280 [ s 2 m ] ⋅ 2 1 = 640 [ s m ]
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