To find the total energy of a body performing harmonic oscillating motion, we need to consider both its kinetic energy (KE) and potential energy (PE).
The kinetic energy of the oscillating body can be calculated using the formula:
KE = (1/2) * m * v^2
where m is the mass of the body and v is the velocity of the body.
The potential energy of the body can be calculated using the formula:
PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position (amplitude).
In harmonic motion, the relationship between displacement, amplitude, and spring constant is given by x = A * sin(2πft), where A is the amplitude, f is the frequency, and t is time.
Given: Mass (m) = 200 g = 0.2 kg Amplitude (A) = 2 cm = 0.02 m Frequency (f) = 5 Hz
To calculate the total energy, we need to determine the velocity and spring constant. The velocity can be found using the formula v = 2πfA, and the spring constant can be calculated using the formula k = (2πf)^2 * m.
Calculations: Velocity (v) = 2π * 5 * 0.02 = 0.628 m/s Spring Constant (k) = (2π * 5)^2 * 0.2 = 98.78 N/m
Now we can calculate the kinetic energy and potential energy:
KE = (1/2) * 0.2 * (0.628)^2 = 0.039 J (rounded to 3 decimal places) PE = (1/2) * 98.78 * (0.02)^2 = 0.019 J (rounded to 3 decimal places)
The total energy is the sum of the kinetic and potential energy:
Total Energy = KE + PE = 0.039 + 0.019 = 0.058 J (rounded to 3 decimal places)
Therefore, the total energy of the body performing harmonic oscillating motion is approximately 0.058 J.