Respuesta :
Answer:
(a) Displacement = - 3.0576 m
(b) Velocity [tex]=-66.48[/tex] m/s
(c)Acceleration = -753.39 m²/s
(d)The phase motion is 26.7 [tex]\pi[/tex].
(e)Frequency =2.5 Hz.
(f)Time period =0.4 s
Explanation:
Given function is
[tex]x= (5.2 m)cos[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
(a)
The displacement includes the parameter t, so,at time t=5.3 s
[tex]x|_{t=5.3}= (5.2 m)cos[ (5\pi \ rad/s)5.3+ \frac\pi5][/tex]
[tex]= (5.2 m)cos[ 26.5\pi+ \frac\pi5][/tex]
=(5.2)(-0.588)m
= - 3.0576 m
(b)
[tex]x= (5.2 m)cos[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
To find the velocity of simple harmonic motion, we need to find out the first order derivative of the function.
[tex]v=\frac{dx}{dt}[/tex]
[tex]=\frac{d}{dt} (5.2 m)cos[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
[tex]= (5.2 m)(-5\pi)sin[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
[tex]= -26\pi sin[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
Now we can plug our value t=5.3 into the above equation
[tex]v= -26\pi sin[ (5\pi \ rad/s)5.3\ s+ \frac\pi5][/tex]
[tex]=-66.48[/tex] m/s
(c)
To find the acceleration of simple harmonic motion, we need to find out the second order derivative of the function.
[tex]v= -26\pi sin[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
[tex]a=\frac{d^2x}{dt^2}[/tex]
[tex]=\frac{dv}{dt}[/tex]
[tex]=\frac{d}{dt}( -26\pi sin[ (5\pi \ rad/s)t+ \frac\pi5])[/tex]
[tex]= -26\pi (5\pi)cos[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
[tex]= -130\pi^2cos[ (5\pi \ rad/s)t+ \frac\pi5][/tex]
Now we can plug our value t=5.3 into the above equation
[tex]a= -130\pi^2cos[ (5\pi \ rad/s)5.3 \ s+ \frac\pi5][/tex]
= -753.39 m²/s
(d)
The general equation of SHM is
[tex]x=x_mcos(\omega t+\phi)[/tex]
[tex]x_m[/tex] is amplitude of the displacement, [tex](\omega t+\phi)[/tex] is phase of motion, [tex]\phi[/tex] is phase constant.
So,
[tex](\omega t+\phi)=5\pi t+\frac\pi5[/tex]
Now plugging t=5.3s
[tex](\omega t+\phi)=5\pi \times 5.3+\frac\pi5[/tex]
=26.7 [tex]\pi[/tex]
The phase motion is 26.7 [tex]\pi[/tex].
The angular frequency [tex]\omega = 5\pi[/tex]
(e)
The relation between angular frequency and frequency is
[tex]\omega =2\pi f[/tex]
[tex]\therefore f=\frac{\omega}{2\pi}[/tex]
[tex]=\frac{5\pi}{2\pi}[/tex]
[tex]=\frac52[/tex]
= 2.5 Hz
Frequency =2.5 Hz.
(f)
The relation between frequency and time period is
[tex]T=\frac1 f[/tex]
[tex]=\frac1{2.5}[/tex]
=0.4 s
Time period =0.4 s
