A particle P of mass 2kg undergoes simple harmonic motion along the x-axis and is attracted towards the origin by a force who's magnitude is numerically equal to 8XN. If it is initially at rest at x=20m, and assuming a general solution of the form A cos wt + B sin wt, find the
I) Position of the particle at anytime
ii) the speed and velocity of the particle at anytime
iii)the magnitude,period and frequency of the vibration
answers
Answer
To find the position, speed, velocity, magnitude, period, and frequency of the vibration for the particle undergoing simple harmonic motion, we can use the given information and equations of motion.
Firstly, let's solve the differential equation for simple harmonic motion:
m * a = -k * x
Here, m is the mass of the particle (2 kg), a is the acceleration, k is the force constant, and x is the position. From the given information, we can deduce that k = 8 N.
By substituting x = 20 into the equation, we get:
2 * a = -8 * 20
a = -80 / 2
a = -40 m/s^2
Since acceleration is the second derivative of position with respect to time (a = d^2x / dt^2), integrating twice will yield the position equation:
a = -ω^2 * x
d^2x / dt^2 = -ω^2 * x
Here, ω is the angular frequency given by ω = 2πf, where f is the frequency.
To solve this differential equation, we can assume a general solution of the form A cos(ωt) + B sin(ωt). Taking the derivatives, we get:
dx / dt = A * (-ω) sin(ωt) + B * ω cos(ωt)
d^2x / dt^2 = A * (-ω)^2 cos(ωt) - B * ω^2 sin(ωt)
Comparing these equations with our differential equation, we can conclude that:
-ω^2 = -40
ω = √40
Now that we have determined ω, we can find A and B:
Given that the particle is initially at rest (v = 0) and x = 20, we can substitute these values into the position equation:
dx / dt = -A * ω * sin(ωt) + B * ω * cos(ωt) = 0
-A * ω * sin(ω * 0) + B * ω * cos(ω * 0) = 0
B = 0
So, our general solution simplifies to:
x = A * cos(ωt)
Now let's address each part of the question:
i) To find the position of the particle at any time, substitute the values of A = 20 and ω = √40 into the position equation:
x = 20 * cos(√40 * t)
ii) To find the speed (v) and velocity (velocity = dx / dt), take the first derivative of the position equation:
v = dx / dt = -√40 * 20 * sin(√40 * t)
iii) To find the magnitude of the vibration, simply take the absolute value of the position equation:
Magnitude = |x| = 20 * |cos(√40 * t)|
To find the period (T) and frequency (f), we can use the formula:
T = 2π / ω
f = 1 / T
Given ω = √40, we can substitute this value into the equations to find the period and frequency.