Energy In Simple Harmonic Motion

or

Step 1: Speed in simple harmonic motion.
The speed in simple harmonic motion is given by .
Step 2: Find the time when speed is maximum.
The speed is maximum when the sine function is equal to 1, i.e., at . Final Answer:

Step 1: Maximum separation occurs when particles are at opposite extremes.
Step 2: This happens when phase difference is .

Step 1: Motion inside the earth is SHM due to linear variation of gravity with radius.
Step 2: Time period equals that of a satellite orbiting close to earth:

Step 1: Maximum kinetic energy: Kmax=(1)/(2)mω²A²
Step 2: Substituting values: 8×10⁻3=(1)/(2)(0.1)ω²(0.1)² ⟹ ω=4rad s⁻1
Step 3: Equation of motion: y=Asin(ω t+φ)=0.1sin(4t+(π)/(4))

Step 1: Total energy in SHM: E = (1)/(2) m ω² a²
Step 2: Average KE from mean to extreme is half of total energy: ⟨ K ⟩ = (E)/(2)
Step 3: ω = 2πν ⟹ ⟨ K ⟩ = π² m a² ν²

Step 1: Simplify the given expression: x = -Asin(2ω t) + Bsin²ω t Using identity sin²ω t = (1-cos 2ω t)/(2), x = -Asin(2ω t) + (B)/(2) - (B)/(2)cos(2ω t)
Step 2: Rearranging, x - (B)/(2) = -Asin(2ω t) - (B)/(2)cos(2ω t) This represents SHM with angular frequency 2ω.
Step 3: Amplitude of SHM: Aₑff = √(A² + ((B)/(2))²)
Step 4: Since the motion repeats after equal intervals of time, it is also periodic.