An object is suspended from a spring and vibrates with simple harmonic motion. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic and what fraction is potential?
Potential energy of a vibrating spring is proportional to the square of the displacement (P.E = kx^2/2, where k is the spriong constant). At maximum displacement, ALL of the energy is potential energy, because motion temporaily stops and KE is zero. At 1/2 maximum displacement, the P.E. is 1/4 of that total energy. That means that 3/4 must be kinetic energyn there.
To determine the fractions of kinetic and potential energy in a simple harmonic motion system, we need to consider the properties of the motion. The total energy of the system remains constant throughout the motion.
In simple harmonic motion, the total mechanical energy is the sum of kinetic energy (KE) and potential energy (PE). At any point during the motion, these two energies will add up to give the total energy (TE) of the system:
TE = KE + PE
Since the system's total energy remains constant, we can calculate the fractions of kinetic and potential energy at a specific instant when the displacement is equal to one-half the amplitude.
Let's assume K represents the fraction of kinetic energy and P represents the fraction of potential energy.
At the extreme points of the motion (maximum displacement), the object's displacement is equal to the amplitude. At these points, the object is momentarily at rest (velocity is zero), and all of the energy is in the form of potential energy. Therefore:
K = 0 (fraction of kinetic energy)
P = 1 (fraction of potential energy)
At the midpoint of the motion, where the displacement is one-half the amplitude, the velocity is maximum, and the potential energy is zero.
To calculate the fractions, we can use the following equation:
K/P = (1/2) * (vmax^2 / (1/2) * (xmax^2)
Where K is the fraction of kinetic energy, P is the fraction of potential energy, vmax is the maximum velocity, and xmax is the maximum displacement (amplitude).
Given that the displacement at this instant is one-half the amplitude, we have:
xmax = (1/2) * amplitude
Now, from the equation of simple harmonic motion, we can find the relationship between velocity and displacement:
vmax^2 = angular frequency^2 * (amplitude^2 - xmax^2)
Substituting the value for xmax:
vmax^2 = angular frequency^2 * (amplitude^2 - (1/2)^2 * amplitude^2)
vmax^2 = angular frequency^2 * (3/4) * amplitude^2
Substituting these values into the equation for K/P:
K/P = (1/2) * (vmax^2 / (1/2) * (xmax^2)
= (1/2) * (angular frequency^2 * (3/4) * amplitude^2 / (1/2) * ((1/2) * amplitude)^2)
= (1/2) * (4/3) * (3/4)
= 1/2
Hence, at an instant when the displacement of the object is equal to one-half the amplitude, half of the total energy of the system is kinetic, and the other half is potential.