The equation of a SHM of a 600 g object is given as x = 0.45 cos 6.4 t where the displacement x is in meters and time in seconds. What is the total energy of the system?
To find the total energy of the system, we need to know the equation for the potential energy (U) and kinetic energy (K) of a simple harmonic oscillator.
In the case of a simple harmonic motion, the potential energy (U) and kinetic energy (K) are related by the equation:
Total energy (E) = Potential energy (U) + Kinetic energy (K)
The equation for potential energy of a simple harmonic oscillator is given by:
U = (1/2) k x^2
where k is the spring constant and x is the displacement.
The equation for kinetic energy of a simple harmonic oscillator is given by:
K = (1/2) m v^2
where m is the mass of the object and v is the velocity.
Now, let's find the values of k, m, x, and v for the given SHM equation: x = 0.45 cos(6.4t)
From the given equation, we can see that the amplitude of the oscillation is 0.45 meters. The amplitude (A) is equal to the maximum displacement (x_max) of the object from its equilibrium position.
Since the maximum displacement is equal to the amplitude, x_max = A = 0.45 meters.
From the equation, we can deduce that the spring constant (k) is related to the angular frequency (ω) as k = mω^2.
The angular frequency (ω) is given by ω = 2πf, where f is the frequency of the oscillation.
In the given equation, the coefficient of t is 6.4, which represents the angular frequency (ω = 6.4 radians/sec).
Now, let's calculate the mass (m) of the object.
The given equation mentions that the object has a mass of 600 grams. To convert it to kilograms, we divide 600 grams by 1000 (since 1 kg = 1000 grams).
m = 600 grams ÷ 1000 = 0.6 kg
Now we have the values of k, m, and x. We can calculate the potential energy (U) and kinetic energy (K) of the system.
U = (1/2) k x^2
U = (1/2) (mω^2) x_max^2
Substituting the known values:
U = (1/2) (0.6 kg) (6.4 rad/s)^2 (0.45 m)^2
Now, let's calculate the kinetic energy (K) of the system.
K = (1/2) m v^2
K = (1/2) (0.6 kg) (v^2)
To find v, we need to differentiate the equation x = 0.45 cos(6.4t) with respect to time.
dx/dt = -0.45 (6.4 sin(6.4t))
v = -0.45 (6.4 sin(6.4t))
Now, let's substitute the known values and calculate K.
K = (1/2) (0.6 kg) (-0.45 (6.4 sin(6.4t)))^2
Finally, we can find the total energy (E) of the system by adding up the potential energy (U) and kinetic energy (K):
E = U + K
Substituting the calculated values, we can find the total energy.