Consider a simple harmonic oscillation with m=0.5kg, k=10N/m and amplitude A=3cm. What is the total energy of the oscillation?

To find the total energy of the simple harmonic oscillation, we need to consider both the potential energy and the kinetic energy.

The potential energy (PE) of the oscillator is given by the equation:

PE = (1/2) * k * x^2

Where k is the spring constant, and x is the displacement from the equilibrium position.

The kinetic energy (KE) of the oscillator is given by the equation:

KE = (1/2) * m * v^2

Where m is the mass of the object, and v is the velocity of the object at any given point during the oscillation.

To find the total energy, we need to sum up the potential and kinetic energy:

Total Energy = PE + KE

Let's calculate it step by step:

First, let's convert the amplitude from centimeters to meters:

A = 3 cm = 0.03 m

Next, let's calculate the potential energy (PE):

PE = (1/2) * k * x^2

Since we have the amplitude (A), which is the maximum displacement from the equilibrium position, we can use it to calculate the displacement, x. For a simple harmonic motion, the displacement is given by:

x = A * cos(ωt)

Where ω is the angular frequency, given by:

ω = sqrt(k / m)

Let's calculate ω:

ω = sqrt(10 N/m / 0.5 kg)
= sqrt(20 rad/s)
≈ 4.47 rad/s

Now, let's calculate the displacement:

x = A * cos(ωt)
= 0.03 m * cos(4.47 t)

Let's substitute the value of x into the potential energy equation:

PE = (1/2) * k * x^2
= (1/2) * 10 N/m * (0.03 m * cos(4.47 t))^2

Next, let's calculate the kinetic energy (KE). We need to calculate the velocity (v) first. The velocity is given by:

v = dx/dt

Let's differentiate the displacement equation with respect to time t to find v:

v = -0.03 m * 4.47 sin(4.47 t)

Now, let's substitute the value of v into the kinetic energy equation:

KE = (1/2) * m * v^2
= (1/2) * 0.5 kg * (-0.03 m * 4.47 sin(4.47 t))^2

Now, we can substitute the values of PE and KE into the total energy equation:

Total Energy = PE + KE

Simply add the expressions for PE and KE:

Total Energy = (1/2) * 10 N/m * (0.03 m * cos(4.47 t))^2 + (1/2) * 0.5 kg * (-0.03 m * 4.47 sin(4.47 t))^2

Simplifying the equation and removing common factors:

Total Energy = 0.015 (cos^2(4.47 t) + sin^2(4.47 t))

According to the trigonometric identity:

cos^2(θ) + sin^2(θ) = 1

So, the total energy of the oscillation is:

Total Energy = 0.015 Joules