a particle of mass 0.3kg vibrates with a period of 2 sec. if its amplitude is 0.5m what is its maximum kinetic energy
Look up the formula that relates maximum speed to amplitude and angular velocity. Then use KE = (1/2) M V^2
Show your work for futher assistance.
3pi/20
Well, I hope this particle is having a wild time on its vibration journey!
To calculate the maximum kinetic energy of the particle, we can use the formula:
K.E. = (1/2) * m * (ω * A)^2
Where:
K.E. is the maximum kinetic energy
m is the mass of the particle (0.3 kg)
ω is the angular frequency (ω = 2π / T, where T is the period)
A is the amplitude of vibration (0.5 m)
Let's plug in the values and find out the answer!
First, let's calculate the angular frequency:
ω = 2π / T = 2π / 2 sec = π rad/sec
Now, we'll use this value to calculate the maximum kinetic energy:
K.E. = (1/2) * 0.3 kg * (π rad/sec * 0.5 m)^2
After some calculations, we get:
K.E. ≈ 0.589 J
So, the maximum kinetic energy of that lively particle is approximately 0.589 Joules. I hope it's having a blast out there!
To determine the maximum kinetic energy of the vibrating particle, we can use the equation for the kinetic energy of a simple harmonic oscillator.
The equation for the kinetic energy (KE) of a vibrating particle is given by:
KE = (1/2) * mass * angular velocity^2 * amplitude^2
To find the angular velocity (ω), we can use the equation:
ω = 2π / T
where T is the period of the motion.
Let's calculate the angular velocity first:
ω = 2π / T
= 2π / 2
= π rad/s
Now we can substitute the values into the kinetic energy equation:
KE = (1/2) * mass * angular velocity^2 * amplitude^2
= (1/2) * 0.3 kg * (π rad/s)^2 * (0.5 m)^2
≈ 0.235 J
Therefore, the maximum kinetic energy of the vibrating particle is approximately 0.235 Joules.
To find the maximum kinetic energy of the particle, we first need to determine its maximum velocity.
The maximum velocity of a particle in simple harmonic motion (SHM) occurs when the displacement is at its maximum value, which is equal to the amplitude of the motion.
In SHM, the displacement of a particle at any time (t) can be represented by the equation:
x = A * sin(2πt / T)
where:
x = displacement
A = amplitude of the motion
t = time
T = period
Given that the amplitude (A) is 0.5m and the period (T) is 2 seconds, we can plug in these values into the equation:
x = 0.5 * sin(2πt / 2)
Simplifying further, we have:
x = 0.5 * sin(πt)
To find the maximum velocity, we need to differentiate the displacement equation with respect to time (t) to get the velocity equation:
v = dx / dt = d(0.5 * sin(πt)) / dt
Differentiating, we have:
v = 0.5π * cos(πt)
The maximum value of cosine (cos(πt)) is 1, so the maximum velocity (v_max) is:
v_max = 0.5π * 1
= 0.5π m/s
Now, to calculate the maximum kinetic energy (K_max) of the particle, we can use the equation:
K_max = (1/2) * m * v_max^2
Given that the mass (m) is 0.3kg, we can substitute the values into the equation:
K_max = (1/2) * 0.3 * (0.5π)^2
Calculating further, we have:
K_max ≈ 0.112 π^2 joules
So, the maximum kinetic energy of the particle is approximately 0.112 π^2 joules.