A particle of mass 4 units moves along x axis attracted towards the origin by a force with magnitude 8x. if it is initially at rest at x=10, then the frequency of the particle is
10
To find the frequency of the particle, we need to determine its equation of motion.
The force acting on the particle is given by:
F = ma = 8x
Where m is the mass of the particle and a is its acceleration.
Since the particle is attracted towards the origin, the force is negative as it opposes the direction of motion.
Since F = ma, we can write:
-ma = 8x
Rearranging the equation, we have:
a = -8x/m
Since a = d²x/dt², we can rewrite the equation as follows:
d²x/dt² = -8x/m
This is a second-order linear differential equation with constant coefficients. The general solution to this equation is of the form:
x(t) = A sin(ωt) + B cos(ωt)
where A and B are constants, and ω is the angular frequency.
To find the frequency, we need to determine the value of ω.
Taking the second derivative of x(t), we have:
d²x/dt² = -Aω² sin(ωt) - Bω² cos(ωt)
Comparing the given equation with the general solution, we can equate the coefficients to get:
ω² = 8/m
Simplifying, we have:
ω = √(8/m)
The frequency is given by f = ω/2π. Substituting the value of ω, we have:
f = √(8/m) / 2π
Given that the mass (m) is 4 units, we can substitute this value into the equation:
f = √(8/4) / 2π
Simplifying, we have:
f = √2 / 2π
Therefore, the frequency of the particle is approximately equal to 0.225/π or 0.071 Hz.
To find the frequency of the particle, we need to first determine its motion equation. Given that the particle is attracted towards the origin by a force with magnitude 8x, we can use Newton's second law to form the differential equation of motion.
Newton's second law states that the force acting on an object is equal to the rate of change of its momentum, which in turn is equal to the mass of the object multiplied by its acceleration. Mathematically, we can represent this as:
F = m * a
In this case, the force acting on the particle is given by 8x, and the mass of the particle is 4 units. Therefore, we have:
8x = 4 * a
To find the motion equation, we need to solve this differential equation. We can do this by rearranging the equation:
a = 2x
This is a second-order linear homogeneous differential equation, which suggests that the general solution will have the form:
x(t) = A * cos(ω * t) + B * sin(ω * t)
Where A and B are constants, t represents time, and ω is the angular frequency.
To determine the values of A and B, we need to use the initial conditions. In this case, the particle is initially at rest at x = 10. This implies that:
x(0) = A * cos(0) + B * sin(0) = 10
Since cos(0) = 1 and sin(0) = 0, we have:
A = 10
Now, to find the angular frequency ω, we can differentiate the position equation with respect to time to obtain the velocity equation:
v(t) = dx(t)/dt = -A * ω * sin(ω * t) + B * ω * cos(ω * t)
We know that the particle is initially at rest, so v(0) = 0:
0 = -10 * ω * sin(0) + B * ω * cos(0)
0 = 0 + B * ω
B * ω = 0
Since ω cannot be zero (as it represents the angular frequency), we have B = 0.
Therefore, the motion equation becomes:
x(t) = 10 * cos(ω * t)
The frequency f of an oscillating motion is given by the reciprocal of the time period T. The time period is the time taken for one complete cycle of the motion. In this case, one complete cycle corresponds to a complete oscillation from the maximum displacement to the same maximum displacement in the opposite direction.
For simple harmonic motion given by x(t) = A * cos(ω * t), the time period T is given by:
T = 2π/ω
Hence, the frequency f is given by the reciprocal of the time period:
f = 1/T = ω/2π
Now, let's find the value of ω for our motion equation:
x(t) = 10 * cos(ω * t)
Comparing this with the general form of simple harmonic motion, we can see that ω = 1.
Finally, substituting the value of ω into the frequency equation, we have:
f = 1/(2π) ≈ 0.159 Hz
Therefore, the frequency of the particle is approximately 0.159 Hz.
since F=ma, a = -8x/4 = -2x
x" + 2x = 0
the general solution to this is
x = c1 sin(√2 t) + c2 cos(√2 t)
so, the frequency is 1/(√2 π)