Solution Found!
Answer: In describing the heat capacities of solids in
Chapter 18, Problem 80P(choose chapter or problem)
Problem 80P
In describing the heat capacities of solids in Section 18.4, we stated that the potential energy a harmonic oscillator averaged over one period of the motion is equal to the kinetic energy
averaged over one period. Prove this result using Eqs. (14.13) and (14.15) for the position and velocity of a simple harmonic oscillator. For simplicity, assume that the initial position rind velocity make the phase angle ϕ up equal to zero. (Hint: Use the trigonometric identities cos2(θ) = [1 + cos(2θ))/2 and sin2(θ) = [1 − cos(2θ))/2. What is the average value of cos(2ωt) overone period?)
Displacement of simple harmonic Oscillator:
Velocity at a Displacement x:
Questions & Answers
QUESTION:
Problem 80P
In describing the heat capacities of solids in Section 18.4, we stated that the potential energy a harmonic oscillator averaged over one period of the motion is equal to the kinetic energy
averaged over one period. Prove this result using Eqs. (14.13) and (14.15) for the position and velocity of a simple harmonic oscillator. For simplicity, assume that the initial position rind velocity make the phase angle ϕ up equal to zero. (Hint: Use the trigonometric identities cos2(θ) = [1 + cos(2θ))/2 and sin2(θ) = [1 − cos(2θ))/2. What is the average value of cos(2ωt) overone period?)
Displacement of simple harmonic Oscillator:
Velocity at a Displacement x:
ANSWER:
Solution 80P
Step 1 of 4:
To calculate the potential energy of the particle,
- The equation of displacement of a particle executing SHM at an instant t is given as:
………...1
Where, A is Amplitude of oscillation
ω = Angular frequency =
Potential energy is given by,
Using equation 1,
Using k=M
………...2