A particle executes simple harmonic motion with an amplitude 4cm. At what displacement it's energy is half kinetic energy and half potential
energy is proportional to amplitude squared.
if KE is 1/2, then
.5KE/KE=(x/4cm)^2
x^2=.5*16
x=4/sqrt2
To find the displacement at which the energy of the particle is half kinetic and half potential, we need to first understand the energy components in simple harmonic motion.
In simple harmonic motion, the total mechanical energy (E) of the particle is the sum of its kinetic energy (KE) and potential energy (PE). The formula for the total mechanical energy is:
E = KE + PE
The kinetic energy of the particle is given by the formula:
KE = (1/2)mv^2
Where m is the mass of the particle and v is its velocity.
The potential energy of the particle is given by the formula:
PE = (1/2)kx^2
Where k is the spring constant and x is the displacement of the particle from its equilibrium position.
In simple harmonic motion, the total mechanical energy is constant throughout the motion, so we can write:
E = constant
Given that the amplitude of the motion is 4 cm, we know that at the extreme points (maximum displacement), the entire energy is either kinetic or potential.
At the extreme points (maximum displacement), all the energy is potential energy. Therefore, we can write:
E = PE_max
Now let's find the values of kinetic and potential energy when the total energy is half kinetic and half potential.
Half the total energy will be:
E/2 = (1/2)PE_max
We can equate this to the potential energy formula to find the displacement at which the energy is half kinetic and half potential:
(1/2)PE_max = (1/2)kx^2
Canceling the common factors, we have:
PE_max = kx^2
Since PE_max and k are constants, we can simplify the equation to:
x^2 = constant
This equation tells us that the displacement (x) at which the energy is half kinetic and half potential is a constant value. In other words, the displacement at which the energy is equally divided between kinetic and potential is independent of the spring constant (k) or the total energy (E).
Therefore, regardless of the values of k and E, the displacement at which the energy is half kinetic and half potential in simple harmonic motion with an amplitude of 4 cm will always remain the same.