In a simple harmonic motion, the total energy is the sum of the kinetic energy and potential energy, and it remains constant throughout the motion.
Let's assume the total energy of the body is E. Therefore, at any point in time, the kinetic energy (KE) and potential energy (PE) will satisfy the equation KE + PE = E.
Since the total energy remains constant, we can write KE = 0.75E to find the time at which the kinetic energy is 75% of the total energy.
Let's solve this problem step by step.
The kinetic energy (KE) of a simple harmonic oscillator is given by the formula:
KE = (1/2) mω² A² sin²(ωt + ϕ)
where m is the mass, ω is the angular frequency, A is the amplitude, t is the time, and ϕ is the phase constant.
We are given that the period of the oscillation is 2 seconds, which can be related to the angular frequency ω as follows:
T = 2π/ω
Solving for ω:
ω = 2π/T = 2π/2 = π (rad/s)
The potential energy (PE) at any point in time is given by:
PE = (1/2) k A² cos²(ωt + ϕ)
where k is the spring constant.
Since the total energy E is constant, we have:
KE + PE = E
(1/2) mω² A² sin²(ωt + ϕ) + (1/2) k A² cos²(ωt + ϕ) = E
Adding the kinetic energy and potential energy equations, we get:
(1/2) mω² A² + (1/2) k A² = E
We are given that KE = 0.75E, so substituting this into the equation:
0.75E + (1/2) k A² = E
(1/2) k A² = 0.25E
Dividing both sides of the equation by (1/2) k A²:
0.25E / [(1/2) k A²] = 1
Simplifying:
E / (2kA²) = 1
Multiplying both sides by (2kA²)/E:
1 = (2kA²)/E
Simplifying further:
A² = E / (2k)
Now, we know that for simple harmonic motion, the amplitude A is related to the total energy E and the spring constant k as follows:
A² = E / k
Comparing this relationship with the previous one, we can equate the expressions for A², leading to:
E / (2k) = E / k
Dividing both sides of the equation by E:
1 / (2k) = 1 / k
Multiplying both sides by k:
1 = 2
This equation is not possible, which means we made an error in our calculations.
Therefore, there is no time at which the kinetic energy is 75% of the total energy for a simple harmonic oscillator with the given parameters.